UDEC JOB TITLE : FIBER-REINFORCED SHOTCRETE LINED TUNNEL. Itasca Consulting Group, Inc. Minneapolis, Minnesota USA

Transcription

1 UDEC Universal Distinct Element Code Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE : FIBER-REINFORCED SHOTCRETE LINED TUNNEL (*10^1) UDEC (Version 5.00) LEGEND Jan 11 12:45 cycle 7152 time 1.757E+00 sec block plot Axial Force on Structure Type # Max. Value struct E+05 structural elements plotted Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) 2011 Itasca Consulting Group Inc. Phone: (1) Mill Place Fax: (1) Third Avenue South, Suite software@itascacg.com Minneapolis, Minnesota USA Web:

2 First Edition January 2000 Second Edition November 2004 First Revision August 2005* Third Edition March 2011 * Please see the errata page in the \Manuals\Udec500 folder.

3 Special Features Structures/Fluid Flow/Thermal/Dynamics 1 PRECIS This volume contains descriptions of some special features of UDEC. The structural element models in UDEC are discussed in Section 1. Four types of structural elements can be used: local reinforcement, cable elements, beam elements and structural supports (props). Modeling of fluid flow in joints is explained in Section 2. Several examples, including thermal coupling, are presented. Section 3 describes the thermal model, and presents several verification problems that illustrate its application with and without mechanical stress. Section 4 contains a description of dynamic modeling in UDEC. Special considerations for running a dynamic analysis are provided, and several verification examples are included.

4 2 Special Features Structures/Fluid Flow/Thermal/Dynamics

5 Special Features Structures/Fluid Flow/Thermal/Dynamics Contents - 1 TABLE OF CONTENTS 1 STRUCTURAL ELEMENTS 1.1 Introduction Reinforcement Local Reinforcement at Joints (REINFORCE Command) Axial Behavior Shear Behavior Numerical Formulation Estimation of Active Length Local Reinforcement Properties Summary of Commands Associated with Local Reinforcement Elements Example Application Reinforced Slope Global Shearing-Resistant Reinforcement (CABLE Command) Axial Behavior Shear Behavior of Grout Annulus Normal Behavior at Grout Interface Cable Element Properties Pretensioning Cable Elements Estimating the Maximum Length for Cable Element Segments Connecting Cable Elements to Beam Elements and to Other Cable Elements Summary of Commands Associated with Cable Elements Example Application Pull-Test for a Grouted Cable Anchor Global Shearing- and Bending-Resistant Reinforcement (STRUCT rockbolt Command) Behavior of Rockbolt Segments Behavior of Shear Coupling Springs Behavior of Normal Coupling Springs Rockbolt-Element Properties Commands Associated with Rockbolt Elements Example Application Rockbolt Pullout Tests Example Application Rockbolt Shear Tests

6 Contents - 2 Special Features Structures/Fluid Flow/Thermal/Dynamics 1.3 Surface Support Structural Beam Elements (STRUCT generate Command) Structural (Beam) Element Formulation Structural Element Generation Structural Element Properties End Conditions and Applied Pressure Summary of Commands Associated with Structural Elements Example Application Inelastic Material Behavior of a Cantilever Beam Example Application Support of a Wedge in a Tunnel Roof Example Application Circular Tunnel Excavation with Interior Support Example Application Shotcrete Lined Tunnel Example Application Slope Stabilization Support Members (SUPPORT Command) Standard Formulation Load-Rate Dependency Numerical Stability Support Member Properties Summary of Commands Associated with Support Members Example Application Support of Faulted Ground Example Application Load-Rate Dependent Support Material Properties Modeling Considerations D/3D Equivalence Symmetry Conditions Equilibrium Conditions Sign Convention References FLUID FLOW IN JOINTS 2.1 Introduction Fluid-Flow Formulations Basic Algorithm Transient Flow of a Compressible Fluid Steady-State Flow Algorithm Transient Flow of an Incompressible Fluid Transient Flow of a Compressible Gas Two-Phase Flow in Joints Modifying Joint Flow Rate and Simulating Viscoplastic Flow in Joints Fluid Boundary Logic One-Way Thermal-Hydraulic Coupling

7 Special Features Structures/Fluid Flow/Thermal/Dynamics Contents Hydraulic Behavior of Rock Joints Parallel Plate Model Fluid Flow Properties and Units Calculation Modes and Commands for Fluid-Flow Analysis Selection of Calculation Mode Properties Boundary Conditions Initialization of Fluid Pressures in Domains Initialization of Pore Pressures in Blocks Solution Output Options Verification and Example Problems Heave of a Rock Layer Example Application of the Fluid Boundary Steady-State Fluid Flow with Free Surface Pressure Distribution in a Fracture with Uniform Permeability (Aperture) Transient Fluid Flow in a Single Joint in an Elastic Medium Transient One-Dimensional Gas Flow Filling of a Horizontal Joint No Capillary Effects Filling of a Horizontal Joint by the Capillary Forces Only Containment of Gas Inside a Cavity Single Horizontal Joint Containment of Gas Inside a Cavern Jointed Rock Mass Thermal-Mechanical-Fluid Flow Example References THERMAL ANALYSIS 3.1 Introduction Formulation Basic Equations Diffusion Equation Explicit Algorithm Stability and Accuracy of the Explicit Scheme Diffusion Equation Implicit Thermal Logic Stability and Accuracy of the Implicit Scheme Thermal-Stress Coupling Solving Thermal-Only and Coupled-Thermal Problems Thermal Analysis Thermal-Mechanical Analysis Heat Transfer across Joints Thermal Boundary Locations

8 Contents - 4 Special Features Structures/Fluid Flow/Thermal/Dynamics 3.4 Input Instructions for Thermal Analysis UDEC Commands FISH Variables Systems of Units for Thermal Analysis Verification Examples Conduction through a Composite Wall Thermal Response of a Heat-Generating Slab Heating of a Hollow Cylinder Infinite Line Heat Source in an Infinite Medium References DYNAMIC ANALYSIS 4.1 Overview Dynamic Formulation Dynamic Modeling Considerations Dynamic Loading and Boundary Conditions Application of Dynamic Input Baseline Correction Quiet Boundaries Free-Field Boundaries Deconvolution and Selection of Dynamic Boundary Conditions Hydrodynamic Pressures Wave Transmission Accurate Wave Propagation Filtering Mechanical Damping Rayleigh Damping Example Application of Rayleigh Damping Guidelines for Selecting Rayleigh Damping Parameters Local Damping for Dynamic Simulations Solving Dynamic Problems General Methodology Illustration of Procedures: Stability of a Jointed-Rock Slope Validation Examples Natural Periods of an Elastic Column Slip Induced by Harmonic Shear Wave Line Source in an Infinite Elastic Medium with a Single Discontinuity References

9 Special Features Structures/Fluid Flow/Thermal/Dynamics Contents - 5 TABLES Table 1.1 Commands associated with local reinforcement elements Table 1.2 Commands associated with cable elements Table 1.3 Commands associated with rockbolt elements Table 1.4 Commands associated with structural elements Table 1.5 Joint set orientations Table 1.6 Support properties for profile props, sticks and yielding props (CSIR 1993) Table 1.7 Commands associated with support members Table 1.8 Systems of units structural elements Table 2.1 Typical SI units for fluid flow parameters Table 2.2 Summary of fluid flow commands Table 2.3 Poroelastic constants for some rocks [Detournay and Cheng 1993] Table 3.1 Summary of thermal commands Table 3.2 System of SI units for thermal problems Table 3.3 System of Imperial units for thermal problems Table 3.4 Problem specifications Table 3.5 Comparison of UDEC results and the analytical solution Table 4.1 Moduli appropriate to various deformation modes Table 4.2 Material properties Table 4.3 Comparison of theoretical and calculated (UDEC) dynamic period T of oscillation for three modes

10 Contents - 6 Special Features Structures/Fluid Flow/Thermal/Dynamics FIGURES Figure 1.1 Axial behavior of local reinforcement systems Figure 1.2 Shear behavior of reinforcement system Figure 1.3 Assumed reinforcement geometry after shear displacement, u s Figure 1.4 Orientation of shear and axial springs representing reinforcement prior to and Figure 1.5 after shear displacement Resolution of reinforcement shear and axial forces into components parallel and perpendicular to discontinuity Figure 1.6 Slope with steeply dipping foliation planes Figure 1.7 Slope failure by reverse toppling Figure 1.8 Stabilization of slope by reinforcement Figure 1.9 Conceptual mechanical representation of fully bonded reinforcement which accounts for shear behavior of the grout annulus Figure 1.10 Cable material behavior for cable elements Figure 1.11 Grout material behavior for cable elements Figure 1.12 Geometry of triangular finite difference zone and transgressing reinforcement used in distinct element formulation Figure 1.13 Axial force and displacement vectors for pull-test Figure 1.14 Cable grout shear force versus displacement at node in small block Figure 1.15 Material behavior of shear coupling spring for rockbolt elements Figure 1.16 Material behavior of normal coupling spring for rockbolt elements Figure 1.17 Rockbolt element in grid; velocity applied at top end node Figure 1.18 Figure 1.19 Figure 1.20 Figure 1.21 Rockbolt pull force (N) versus rockbolt axial displacement (meters) for a single 25 mm grouted rockbolt Rockbolt pull force (N) versus rockbolt axial displacement (meters) for a single 25 mm grouted rockbolt with displacement weakening Rockbolt pull force (N) versus rockbolt axial displacement (meters) for a single 25 mm grouted rockbolt with 5 MPa in-plane confinement and zero outof-plane confinement Rockbolt pull force (N) versus rockbolt axial displacement (meters) for a single 25 mm grouted rockbolt with 5 MPa in-plane confinement plus a reduction factor and zero out-of-plane confinement Figure 1.22 Rockbolt pull force (N) versus rockbolt axial displacement (meters) for a single 25 mm grouted rockbolt with tensile rupture Figure 1.23 Rockbolt shear force (N) versus rockbolt shear displacement (meters) for a single 25 mm grouted rockbolt Figure 1.24 Deformed shape of 25 mm diameter rockbolt at end of shear test Figure 1.25 Figure 1.26 Rockbolt shear force (N) versus rockbolt shear displacement (meters) for a single 25 mm grouted rockbolt with tensile rupture Deformed shape of 25 mm diameter rockbolt following rupture at end of shear test

11 Special Features Structures/Fluid Flow/Thermal/Dynamics Contents - 7 Figure 1.27 Local stiffness matrix for structural element representation of excavation support Figure 1.28 Lumped mass representation of structure used in explicit formulation Figure 1.29 Demonstration of interface slip and large displacement capabilities of explicit structural element formulation Figure 1.30 Typical moment-thrust diagram Figure 1.31 Parameters to define structural element locations Figure 1.32 Shape factors and inertial moments for different shapes Figure 1.33 Moment-thrust diagram at initial compressive and tensile strengths Figure 1.34 Moment-thrust diagram include residual compressive and tensile strengths 1-79 Figure 1.35 Moment-thrust diagram with zero tensile strength and crack depth ratio = Figure 1.36 Moment-thrust diagram input P-M diagram table Figure 1.37 Lined tunnel with wedge in roof Figure 1.38 Case 1: elastic solution Figure 1.39 Case 2: yield strength = residual strength = 18 MPa Figure 1.40 Case 3: yield strength = 18 MPa, residual strength = 16.7 MPa Figure 1.41 Case 4: two layers of lining Figure 1.42 Case 4: elastic solution for two layers of lining Figure 1.43 Conceptual representation of support reaction and ground reaction curves Figure 1.44 Zoning for UDEC model of circular tunnel excavation Figure 1.45 Comparison of ground reaction/support reaction lines Figure 1.46 Shotcrete applied in 300 arc on tunnel periphery Figure 1.47 Figure 1.48 Figure 1.49 Figure 1.50 Axial force distribution in shotcrete lining residual yield strength = 20 MPa Tensile failure locations in shotcrete residual yield strength = 20 MPa Moment-thrust diagram for tensile yield strength = 20 MPa and compressive yield strength = 40 MPa Axial force distribution in shotcrete lining residual yield strength = 10 MPa Figure 1.51 Slope cut in a jointed rock Figure 1.52 Unsupported slope is unstable Figure 1.53 Slope is stabilized with shotcrete lining Figure 1.54 Force-displacement behavior for standard support model Figure 1.55 Force-displacement behavior including load-rate dependent support force Figure 1.56 Support members before loading Figure 1.57 Force-displacement relation for support in example problem Figure 1.58 Support members after loading Figure 1.59 Model test for rate-dependent support members Figure 1.60 Axial force versus displacement response for left prop

12 Contents - 8 Special Features Structures/Fluid Flow/Thermal/Dynamics Figure 1.61 Axial force versus displacement response for right prop unloading Figure 1.62 Axial force versus displacement response in left prop for right prop unloading Figure 1.63 Actual axial forces in vertically loaded rockbolt at 2 m spacing (spacing given) Figure 1.64 Scaled axial forces in vertically loaded rockbolt at 2 m spacing (spacing not given) Figure 2.1 Fluid/solid interaction in discontinua Figure 2.2 Flow in joints modeled as flow between domains Figure 2.3 Relation between hydraulic aperture, a, and joint normal stress, σ n,inudec 2-5 Figure 2.4 Extreme example in which the speed of propagation depends on system stiffness Figure 2.5 Capillary pressure curve as a function of saturation and parameter β Figure 2.6 Flow-gradient relation for Newtonian fluid in UDEC Figure 2.7 Flow-gradient relation for Bingham fluid in UDEC Figure 2.8 Porous medium mesh Figure 2.9 Elements in porous medium mesh Figure 2.10 Decomposition of stresses acting on a porous, elastic rock Figure 2.11 Heave of a rock layer Figure 2.12 UDEC model for fluid boundary example Figure 2.13 Fluid pressure field Figure 2.14 Figure showing definition of terms in Dupuit s formula Figure 2.15 UDEC problem geometry for verification of fluid flow logic Figure 2.16 UDEC steady-state flow rates Figure 2.17 UDEC domain pressures Figure 2.18 Model geometry and boundary conditions Figure 2.19 Comparison of analytical and UDEC solutions for joint fluid pressure (P / P o ) at various distances (x/l) in a fracture with zero initial pressure at x = Figure 2.20 UDEC model for incompressible flow in a single joint Figure 2.21 Fluid pressure histories at A, B and C for a 1-second fluid timestep Figure 2.22 Fluid pressure histories at A, B and C for a 10-second fluid timestep Figure 2.23 Fluid pressure histories at A, B and C for a 10-second fluid timestep; run to steady-state flow Figure 2.24 Fluid pressure histories at A, B and C for compressible flow Figure 2.25 Fluid pressure histories at A, B and C for compressible flow; run to steadystate Figure 2.26 UDEC model for 1D gas flow Figure 2.27 Gas pressure along joint ( versus ξ) at time = Figure 2.28 Gas pressure along joint ( versus ξ) at time = Figure 2.29 Geometry of the UDEC model Figure 2.30 Wetting fluid pressure in the joint after 1.43 s Figure 2.31 Saturation of the joint after 1.43 s

13 Special Features Structures/Fluid Flow/Thermal/Dynamics Contents - 9 Figure 2.32 Wetting fluid pressure histories at the points along the joint Figure 2.33 Saturation histories at the points along the joint Figure 2.34 Location of fluid front (distance from the left boundary in meters) as a function of time (seconds). Comparison of the UDEC solution (crosses) with analytical solution (line) Figure 2.35 Wetting fluid pressures along the joint after.014 s Figure 2.36 Saturation along the joint after.014 s Figure 2.37 Wetting fluid pressure histories at the points along the joint Figure 2.38 Saturation histories at the points along the joint Figure 2.39 Geometry of the model Figure 2.40 Wetting fluid pressure along the joint after s Figure 2.41 Saturation along the joint after s Figure 2.42 Model displacements after s Figure 2.43 Non-wetting fluid pressure histories at the points along the joint Figure 2.44 Saturation at the points along the joint Figure 2.45 Geometry of the model Figure 2.46 Non-wetting fluid pressures after 0.25 s Figure 2.47 Saturation of the joints in the model after 0.25 s Figure 2.48 UDEC model for thermal-mechanical-fluid flow example Figure 2.49 Fluid pressure versus thermal time histories for transient flow analysis Figure 2.50 Fluid pressure versus thermal time histories for steady-state flow analysis Figure 3.1 Heat flow into gridpoint k Figure 3.2 General solution procedure for thermal-mechanical analysis Figure 3.3 Subdivision of zones at contacts Figure 3.4 Typical zoning of rigid block Figure 3.5 Zones that may cause inaccuracy Figure 3.6 Composite wall Figure 3.7 Idealization of the wall for the UDEC model Figure 3.8 Zone distribution Figure 3.9 Steady-state temperature distribution Figure 3.10 Temperature vs distance comparison between UDEC and analytical solution 3-30 Figure 3.11 Heat-generating slab showing initial and boundary conditions Figure 3.12 Model conditions for heat-generating slab Figure 3.13 UDEC zone distribution Figure 3.14 Temperature evolution in the center of the slab Figure 3.15 Temperature distribution at steady-state Figure 3.16 UDEC and analytical temperature distributions at thermal time = 0.1, 0.5 and 5.0 seconds (analytical values = odd-numbered tables; numerical values = even-numbered tables) Figure 3.17 UDEC grid for heating of a hollow cylinder Figure 3.18 Temperature distribution at steady state for heating of a hollow cylinder

14 Contents - 10 Special Features Structures/Fluid Flow/Thermal/Dynamics Figure 3.19 Radial stress distribution at steady state for heating of a hollow cylinder Figure 3.20 Tangential stress distribution at steady state for heating of a hollow cylinder 3-47 Figure 3.21 Axial stress distribution at steady state for heating of a hollow cylinder Figure 3.22 UDEC grid for an infinite line heat source Figure 3.23 Close-up view of zoning in blocks Figure 3.24 Temperature distribution at 1 year Figure 3.25 Radial displacement distribution at 1 year Figure 3.26 Radial and tangential stress distributions at 1 year Figure 4.1 Types of dynamic loading and boundary conditions in UDEC Figure 4.2 Primary and reflected waves in a bar: stress input through a quiet boundary 4-7 Figure 4.3 The baseline correction process Figure 4.4 Model for seismic analysis of surface structures and free-field mesh Figure 4.5 x-velocity histories at top of model with free-field boundaries Figure 4.6 Seismic input to UDEC Figure 4.7 Layered system analyzed by SHAKE (layer properties are shear modulus, G, density, ρ, and damping fraction, ζ ) Figure 4.8 Deconvolution procedure for a rigid base (after Mejia and Dawson 2006) Figure 4.9 Deconvolution procedure for a compliant base (after Mejia and Dawson 2006) Figure 4.10 Compliant-base deconvolution procedure for a typical case (after Mejia and Dawson 2006) Figure 4.11 Compliant-base deconvolution procedure for another typical case (after Mejia and Dawson 2006) Figure 4.12 Embankment analyzed with a rigid and compliant base Figure 4.13 (after Mejia and Dawson 2006) Computed accelerations at crest of embankment (after Mejia and Dawson 2006) Figure 4.14 Hydrodynamic pressure acting on a rigid dam with a vertical upstream face 4-21 Figure 4.15 Dam model with hydrodynamic pressure boundary on upstream face Figure 4.16 Comparison of x-displacement at top of dam Figure 4.17 Column of variable-sized blocks subjected to triangular-shaped impulse load Figure 4.18 at base... Input wave (solid) at base and calculated wave (dashed) at top of column of 4-30 rigid block model Figure 4.19 Column of variable-sized blocks subdivided into finite difference zones Figure 4.20 Input wave (solid) at base and calculated wave (dashed) at top of column of deformable block model Figure 4.21 Unfiltered velocity history Figure 4.22 Unfiltered power spectral density plot Figure 4.23 Filtered velocity history at 15 Hz Figure 4.24 Results of filtering at 15 Hz

15 Special Features Structures/Fluid Flow/Thermal/Dynamics Contents - 11 Figure 4.25 Variation of normalized critical damping ratio with angular frequency Figure 4.26 Plot of vertical displacement versus time, for a single block contacting on a rigid base with gravity suddenly applied (no damping) Figure 4.27 Plot of vertical displacement versus time, for a single block contacting on a rigid base with gravity suddenly applied (mass and stiffness damping) Figure 4.28 Plot of vertical displacement versus time, for a single block contacting on a rigid base with gravity suddenly applied (mass damping only) Figure 4.29 Plot of vertical displacement versus time, for a single block contacting on a rigid base with gravity suddenly applied (stiffness damping only) Figure 4.30 Plot of velocity spectrum versus frequency Figure 4.31 Comparison of fundamental wavelengths for bars with varying end conditions 4-46 Figure 4.32 Displacement history 5% Rayleigh damping Figure 4.33 Displacement history 5% local damping Figure 4.34 Initial equilibrium of slope cut in jointed rock Figure 4.35 Input velocity (history 2) and calculated velocity (history 3) at top of model without slope Figure 4.36 Slope failure resulting from dynamic loading Figure 4.37 x-velocity histories at base, slope face and remote from slope Figure 4.38 Transmission and reflection of incident harmonic wave at a discontinuity Figure 4.39 Problem geometry and boundary conditions for the problem of slip induced by harmonic shear wave Figure 4.40 Time variation of shear stress at points A and B for elastic discontinuity (cohesion = 2.5 MPa) Figure 4.41 Time variation of shear stress at points A and B for slipping discontinuity (cohesion = 0.5 MPa) Figure 4.42 Time variation of shear stress at points A and B for slipping discontinuity (cohesion = 0.1 MPa) Figure 4.43 Time variation of shear stress at points A and B for slipping discontinuity (cohesion = 0.02 MPa) Figure 4.44 Comparison of transmission, reflection and absorption coefficients; points denote UDEC results Figure 4.45 Problem geometry for an explosive source near a slip-prone discontinuity Figure 4.46 Dimensionless analytical results for slip history at point P (Day 1985) (dimensionless slip = (4h ρβ 2 /m o )δu; dimensionless time = tβ/h) Figure 4.47 Problem geometry and boundary conditions for the UDEC analysis Figure 4.48 UDEC model showing semicircular source and joints Figure 4.49 Input radial velocity time history prescribed at r = 0.05 h (dimensionless velocity = (h 2 ρβ/m o )v; dimensionless time = tβ/h) Figure 4.50 Comparison of analytical results (history 4) and numerical results (history 8) for dynamic slip at point P, using the Coulomb joint model Figure 4.51 Comparison of analytical results (history 4) and numerical results (history 8) for dynamic slip at point P, using the continuously yielding model

16 Contents - 12 Special Features Structures/Fluid Flow/Thermal/Dynamics EXAMPLES Example 1.1 Reinforced slope Example 1.2 Simulation of a pull-test for a grouted cable anchor Example 1.3 Rockbolt pullout tests Example 1.4 Rockbolt shear tests Example 1.5 Inelastic material behavior of a cantilever beam Example 1.6 Lined tunnel with wedge in roof Example 1.7 Circular excavation with interior support Example 1.8 Shotcrete lined tunnel Example 1.9 Slope stabilization Example 1.10 Support of faulted ground Example 1.11 Load-rate dependent support Example 1.12 Axial loading of rockbolts at 2 m spacing Example 2.1 Heave of a rock layer Example 2.2 Application of the fluid boundary logic Example 2.3 Steady-state fluid flow with a free surface Example 2.4 Pressure distribution in a fracture with uniform permeability Example 2.5 Incompressible transient flow in a single joint in an elastic medium Example 2.6 Compressible transient flow in a single joint in an elastic medium Example 2.7 Analytical solution and empirical fit for 1D gas flow Example 2.8 Analytical solution for pressure at x = Example 2.9 Transient one-dimensional gas flow Example 2.10 Injection of water in gas-filled joint Example 2.11 Filling of the joint with water due to capillary forces only Example 2.12 Containment of gas inside a cavity single joint Example 2.13 Containment of gas inside a cavity jointed rock mass Example 2.14 Thermo-mechanical-fluid flow with incompressible transient flow Example 2.15 Thermal-mechanical-fluid flow assuming steady-state flow Example 3.1 Conduction through a composite wall Example 3.2 Thermal response of a heat-generating slab Example 3.3 Heating of a hollow cylinder Example 3.4 Infinite line heat source in an infinite medium Example 3.5 Exponential integral function Example 4.1 Shear wave propagation in a vertical column Example 4.2 Shear wave loading of a model with free-field boundaries Example 4.3 Hydrodynamic pressure acting on a dam Example 4.4 Column of variable-sized blocks subjected to impulse load at base Example 4.5 Block under gravity undamped and 3 critically damped cases Example 4.6 Continuation of Example 4.5 with 5% Rayleigh damping

17 Special Features Structures/Fluid Flow/Thermal/Dynamics Contents - 13 Example 4.7 Continuation of Example 4.5 with 5% local damping Example 4.8 Initial conditions for the slope problem Example 4.9 Dynamic excitation of the slope problem Example 4.10 Data file for confined compression Example 4.11 Data file for unconfined compression Example 4.12 Data file for shear Example 4.13 Listing of avper.fis : function to compute average period Example 4.14 Verification of dynamic slip four complete simulations Example 4.15 Line source in an infinite elastic medium with a discontinuity Example 4.16 Listing of VEL INP.FIS : function to calculate velocity input Example 4.17 Listing of ANA SLP.FIS : function to calculate Day solution for dynamic slip

18 Contents - 14 Special Features Structures/Fluid Flow/Thermal/Dynamics

19 STRUCTURAL ELEMENTS STRUCTURAL ELEMENTS 1.1 Introduction An important aspect of geomechanical analysis and design is the use of structural support to stabilize a soil or rock mass. The term support describes engineered materials used to restrict displacements in the immediate vicinity of an opening or excavation. In this section, support is divided into two types: reinforcement and surface support. Reinforcement consists of tendons (i.e., cables) or bolts installed in holes drilled in the rock mass. Reinforcement acts to conserve inherent rock mass strength so that it becomes self-supporting. Two types of reinforcement model are provided in UDEC: local and global reinforcement. A local reinforcement model considers only the local effect of reinforcement where it passes through existing discontinuities. This reinforcement type is accessed with the REINFORCE command; the model is described in Section A global reinforcement model considers the presence of the reinforcement along its entire length throughout the rock mass. Two types of global reinforcement are provided in UDEC. One type, cable reinforcement, only provides shear resistance along the cable. This model is accessed via the CABLE command and is described in Section The other type, rockbolt reinforcement, provides shear resistance along the length of the rockbolt, and resistance normal to the length of the bolt including bending resistance. The model is accessed with the STRUCT rockbolt command and is described in Section Surface support consists of concrete lining, steel sets, shotcrete, etc. that are placed on the surface of an excavation and, in many cases, act to truly support, in whole or part, the weights of individual blocks isolated by discontinuities or zones of loosened rock. Structural beam elements are created in UDEC, via the STRUCT generate command, to simulate surface support for both interior excavations (such as tunnels) and surface excavations (such as open cuts and natural slopes). Structural beam support is described in Section Surface support also includes one-dimensional support members that represent hydraulic or wooden props or packs. This support acts as a single degree-of-freedom member that is connected between two boundaries of an interior opening. The SUPPORT command accesses this logic. See Section A general guide to material properties for structural elements is given in Section 1.4, and specific modeling considerations when using structural elements are discussed in Section 1.5. In particular, note that, because UDEC is a two-dimensional program, the three-dimensional effect of regularly spaced elements is accommodated by scaling their material properties in the out-of-plane direction. The scaling is accomplished by assigning spacing as a property for the structural elements. This procedure is explained in Section

20 1-2 Special Features Structures/Fluid Flow/Thermal/Dynamics In all cases, the commands necessary to define each of the structural element types are quite simple, but they invoke a very powerful and flexible structural logic. This logic is developed with the same finite-difference algorithms as the rest of the code (as opposed to a matrix-solution approach), allowing the structure to accommodate large displacements and to be applied for dynamic as well as static analysis. Example applications for each of the structural support types are provided at the end of the section describing each type.* * The data files for the examples given in this section have the extension.dat and are stored in the directory ITASCA\UDEC500\Datafiles\Structures. A project file is also provided for each example. In order to run an example and compare the results to plots in this section, open a project file in the GIIC by clicking on the File / Open Project menu item and selecting the project file name (with extension.prj ). Click on the Project Options icon at the top of the Project Tree Record, select Rebuild unsaved states, and the example data file will be run, and plots created.

21 STRUCTURAL ELEMENTS Reinforcement There are several different types of reinforcement designed to operate effectively in a range of ground conditions. One type is represented by a reinforcing bar or bolt fully encapsulated in a strong, stiff resin or grout. This system is characterized by the relatively large axial resistance to extensions that can be developed over a relatively short length of the shank of the bolt, and by the high resistance to shear that can be developed by an element penetrating a slipping joint. A second type of reinforcement system, represented by cement-grouted cables or rockbolts, offers little resistance to joint shear, and development of full-axial load may require deformation of the grout over a substantial length of the reinforcing element. These two types of reinforcement are identified, respectively, as local reinforcement and global (or spatially extensive) reinforcement. The characteristic behavior of these two rock reinforcement systems has been incorporated into UDEC. Local reinforcement can be applied to both rigid and deformable blocks. Global reinforcement can only be applied to deformable blocks. Both reinforcement systems simulate a row of equally spaced reinforcement in the out-of-plane direction.* Local Reinforcement at Joints (REINFORCE Command) The local reinforcement formulation considers only the local effect of reinforcement where it passes through existing discontinuities. The formulation results from observations of laboratory tests of fully grouted untensioned reinforcement in good quality rocks with one discontinuity, which indicate that strains in the reinforcement are concentrated across the discontinuity (Bjurstrom 1974, and Pells 1974). This behavior can be achieved in the computational model by calculating, for each element, the forces generated by displacements across the discontinuity through which the element passes. The following description of this formulation is taken from Lorig (1985). This formulation exploits simple force-displacement relations to describe both the shear and axial behavior of reinforcement across discontinuities. Large shear displacements are accommodated by considering the simple geometric changes that develop locally in the reinforcement near a discontinuity. Although the local reinforcement model can be used with either rigid blocks or deformable blocks, the representation is most applicable to cases in which deformation of individual rock blocks may be neglected in comparison with deformation of the reinforcing system. In such cases, attention may be reasonably focused on the effect of reinforcement near discontinuities. * Note that reinforcement elements cannot be used to simulate a single vertical element because the structural element formulation does not apply to axisymmetric geometry.

22 1-4 Special Features Structures/Fluid Flow/Thermal/Dynamics Axial Behavior Historically, axial testing of rock reinforcement has focused on pull-out tests for two reasons: (1) ease of experimentation and interpretation of results; and (2) provision of axial restraint (the main function of reinforcement in the prevailing conceptual models). Consequently, a relatively good understanding of axial force-displacement relations has been achieved. The axial force-displacement relation typically used in the representation of rock reinforcement is shown in Figure 1.1: tension P ult rupture 1 K a axial displacement load reversal Figure 1.1 Axial behavior of local reinforcement systems Figure 1.1 indicates an identical response in tension and compression. This may not be the case for all reinforcing systems. If pull-test results are not available, the following theoretical expression given by Gerdeen et al. (1977) may be used to estimate the axial stiffness, K a, for fully bonded solid reinforcing elements: where d 1 = reinforcement diameter; K a = πkd 1 (1.1)

23 STRUCTURAL ELEMENTS 1-5 k =[ 1 2 G g E b /(d 2 /d 1 1)] 1/2 ; G g = grout shear modulus; E b = Young s modulus of reinforcement material; and d 2 = hole diameter. Comparisons with finite element analyses (Gerdeen et al. 1977) indicate that Eq. (1.1) tends to slightly overestimate axial stiffnesses. The ultimate axial capacity of the reinforcement depends on a number of factors, including strength of the reinforcing element, bond strength, hole roughness, grout strength, rock strength and hole diameter. In the absence of results of physical tests, empirical relations may be used to estimate the ultimate anchorage strength, P ult. One such relation for the design of cement-grouted reinforcement is given by Littlejohn and Bruce (1975): P ult = 0.1 σ c πd 2 L (1.2) where σ c = uniaxial compressive strength of massive rocks (100% core recovery) up to a maximum value of 42 MPa, assuming that the compressive strength of the cement grout is equal to or greater than 42 MPa; and L = bond length Shear Behavior Recognition that reinforcement also acts to modify the shear stiffness and strength of discontinuities has led to laboratory shear testing of reinforced discontinuities. Experimental results and theoretical investigations indicate that shearing along a discontinuity induces bending stresses in the reinforcement that decay very rapidly with distance into the rock from the shear surface. Typically, within one to two reinforcing element diameters, the bending stresses are insignificant. The shear force-displacement relation typically used to represent shear behavior is shown in Figure 1.2. The figure shows representative responses for reinforcement at various attitudes with respect to the traversed discontinuity and direction of shear. If the results of physical tests are not available, the shear stiffness, K s, may be estimated using the following expression from Gerdeen et al. (1977). K s = E b Iβ 3 (1.3) where β =[K/(4E b I)] 1/4 ; K =2E g /(d 2 /d 1 1); I = second moment of area of the reinforcement element; and E g =Young s modulus of the grout.

24 1-6 Special Features Structures/Fluid Flow/Thermal/Dynamics shear force θ 0 K s 1 θ 0 = 45 rupture θ 0 = 90 θ 0 = 135 shear displacement Figure 1.2 Shear behavior of reinforcement system Empirical relations can be used to estimate the maximum shear force, Fs,b max, for a reinforcement element at various orientations with respect to a transgressed discontinuity and direction of shear. For example, Bjurstrom (1974) used the results of shear tests of ungrouted reinforcement perpendicular to a discontinuity in granite to develop the expression where σ b = yield strength of reinforcement. F max s,b = 0.67 d 2 1 (σ b σ c ) 1/2 (1.4) In their assessment of maximum shear resistance, St. John and Van Dillen (1983) applied the results of Azuar et al. (1979). The latter found that the maximum shear force was about half the product of the uniaxial tensile strength of the reinforcement and its cross-sectional area for reinforcement perpendicular to the discontinuity. The force increased to 80-90% of that product for reinforcement, inclined with the direction of shear. Shear displacements causing rupture were reported after approximately two reinforcement diameters for the perpendicular case, and one diameter for the inclined case. St. John and Van Dillen interpreted differences between strength and amount of displacement before rupture in terms of the extent of crushing of rock around the reinforcement.

25 STRUCTURAL ELEMENTS Numerical Formulation The model in UDEC assumes that, during shear displacement along a discontinuity, the reinforcement deforms as shown in Figure 1.3. The short length of reinforcement, which spans the discontinuity and changes orientation during shear displacement, is referred to as the active length. The assumed geometric changes were originally suggested in a derivation by Haas (1976) for conventional point-anchored reinforcement and adopted by Fuller and Cox (1978) in considering fully grouted reinforcement. Direction of Shearing 0 Discontinuity Active Length u s (Positive) Figure 1.3 Assumed reinforcement geometry after shear displacement, u s It is assumed that the active length changes orientation only as a direct geometric result of shear and normal displacements at the discontinuity. Methods for estimating the active length are presented in the next section. The model may be considered to consist of two springs located at the discontinuity interface and oriented parallel and perpendicular to the reinforcement axis, as shown in Figure 1.4. Following shear displacement, the axial spring is oriented parallel to the active length, while the shear spring remains perpendicular to the original orientation, as shown in Figure 1.4. Similar geometric changes follow displacements normal to the discontinuity.

26 1-8 Special Features Structures/Fluid Flow/Thermal/Dynamics Axial Spring Discontinuity Shear Spring Direction of Shearing Axial Spring Discontinuity Shear Spring Figure 1.4 Orientation of shear and axial springs representing reinforcement prior to and after shear displacement The force-displacement models used in UDEC to represent axial and shear behavior are continuous, nonlinear algorithms written in terms of stiffness (axial or shear), the ultimate load capacity and a yield function. The yield function describes the force-displacement path followed in approaching the ultimate capacity. The force-displacement relation that describes the axial response is given by the equation F a = K a u a f(f a ) (1.5)

27 STRUCTURAL ELEMENTS 1-9 where F a is an incremental change in axial force; u a is an incremental change in axial displacement; K a is the axial stiffness; and f(f a ) is a function describing the path by which the axial force, F a approaches the ultimate (or bounding) axial force Fa,b max. The function [ f(f a ) = Fa,b max (F max ] a,b F a ) ea F a [Fa,b max (1.6) ]2 is used to represent the axial yield curve. From Eq. (1.5), the axial force seeks the bounding force in an asymptomatic manner. The axial stiffness exponent, e a, controls the rate at which the bounding force is reached. If e a = 0, then the axial stiffness remains constant. In computing the incremental axial displacement of the active length, it is necessary to account for crushing of the grout and/or rock near the discontinuity as shear displacement causes the reinforcement to bear against one side of the hole. In the present model, a reduction factor, r f, is applied to incremental axial displacements arising from changes in orientation of the active length to account for the crushing. The reduction factor is computed from the expression r f = u axial (u 2 s + u2 n ) 1/2 (1.7) where u axial = summation of axial displacement increments (i.e., discontinuity displacement increments resolved at each configuration in the direction of the active length); u s = total discontinuity shear displacement; and = total discontinuity normal displacement. u n Note that no reduction (r f = 1.0) is applied for cases in which there is no change in orientation of the active length. The shear force-displacement relation is described in incremental form by the expression F s = K s u s f(f s ) (1.8) where F s is an incremental change in shear force; u s is an incremental change in shear displacement; K s is the shear stiffness; and f(f s ) is a function describing the path by which the shear force, F s, approaches the ultimate or bounding shear force, Fs,b max.

28 1-10 Special Features Structures/Fluid Flow/Thermal/Dynamics The function [ f(f s ) = Fs,b max (F max ] s,b F s ) es F s [Fs,b max (1.9) ]2 is used to represent the shear yield curve. From Eq. (1.8), the shear force seeks the bounding force in an asymptotic manner. The shear stiffness exponent, e s, controls the rate at which the bounding force is reached. If e s = 0, then the shear stiffness remains constant. The maximum shear force, Fs,b max, changes for various orientations of the active length. The following equation is used to adjust the maximum shear force. where Fs max u s F max s,b = F s max [ 1 + [sign(cos θ0, u s ) cos(θ 0 )] ] = πd1 2 σ b/4; and = incremental change in shear displacement. 2 (1.10) The term sign(cos θ 0, u s ) assigns the sign of u s to cos(θ 0 ). The maximum shear force, Fs,b max, decreases from a maximum at θ 0 = 0 to 50% of Fs max at θ 0 = 90 (see Figure 1.2), which is consistent with the results of Azuar et al. (1979). The force-displacement relations described above are used to determine forces arising in the springs from incremental displacements at the end points of the active length. The resultant shear and axial forces are resolved into components parallel and perpendicular to the discontinuity, as shown in Figure 1.5. Forces are then applied to the neighboring blocks.

29 STRUCTURAL ELEMENTS 1-11 Direction of Shearing Shear Displacement of Reinforcement Discontinuity Resultant Shear Force θ 0 θ 0 Normal Force Applied to Upper Block Shear Force Applied to Upper Block Direction of Shearing Discontinuity Resultant Axial Force Normal Force Applied to Upper Block Shear Force Applied to Upper Block θ Figure 1.5 Resolution of reinforcement shear and axial forces into components parallel and perpendicular to discontinuity

30 1-12 Special Features Structures/Fluid Flow/Thermal/Dynamics Estimation of Active Length An estimate of the active length is required to define the assumed local deformation illustrated in Figure 1.3. It has been shown that the active length extends approximately one to two reinforcing element diameters on either side of the discontinuity. In the absence of experimental data, results of theoretical analysis may be used to define the active length. For example, in defining the elastic shear stiffness, K s, Gerdeen et al. (1977) also determine a quantity, l, called the load transfer length, or decay length. If ρ max is the proportion of maximum deflection in the reinforcement, the relation between it and the load transfer length may be expressed by e βl = ρ max (1.11) For example, the point at which the deflection decays to 5% of its maximum value is or e βl = 0.05 l = 3/β. This approach was developed for reinforcement oriented perpendicular to the shear plane. Dight (1982) presents a theoretical analysis for determining the distance from the shear plane to maximum moment which corresponds with the location of the plastic hinge in the reinforcement element. This approach places no restrictions on the orientation of the reinforcement with respect to the shear plane. A significant result of this analysis is that the distance of the plastic hinge from the shear plane does not appear to vary greatly with shear displacement, especially for displacements greater than 10 mm (0.4 in) for typical reinforcement systems. This observation is in agreement with the assumed geometry changes described earlier Local Reinforcement Properties The local reinforcement elements used in UDEC require several input parameters: (1) axial stiffness [force/length]; (2) axial stiffness exponent; (3) ultimate axial capacity [force]; (4) axial failure strain; (5) shear stiffness [force/length]; (6) shear stiffness exponent;

31 STRUCTURAL ELEMENTS 1-13 (7) ultimate shear capacity [force]; (8) 1/2 active length; (9) reversal factor; and (10) spacing. The axial stiffness, axial stiffness exponent and ultimate axial capacity are usually determined to best fit pull-out tests, as described in Section By default, the axial stiffness exponent is zero, so the axial force-displacement relation follows a constant axial stiffness until the ultimate axial capacity is reached. A limiting axial strain can also be defined; if not specified, the axial strain is unlimited. The shear stiffness, shear stiffness exponent, ultimate shear capacity and 1/2 the active length can also be back-calculated from experimental testing, as discussed in Sections and By default, the shear stiffness exponent is zero, and a constant shear stiffness is assumed until the ultimate shear capacity is reached. A reversal factor can also be specified. This controls the slope of the shear force-displacement relation when the direction of the shear force is reversed. Reasonable values for the reversal factor vary between 0 and 1. If the factor equals 1, then the slope upon reversal is the same as that upon initial loading Summary of Commands Associated with Local Reinforcement Elements All of the commands associated with local reinforcement elements are listed in Table 1.1. See Section 1 in the Command Reference for a detailed explanation of these commands.

32 1-14 Special Features Structures/Fluid Flow/Thermal/Dynamics Table 1.1 Commands associated with local reinforcement elements REINFORCE mat x1 y1 x2 y2 REINFORCE delete <range> PROPERTY mat mat keyword r aexp value r astiff value r length value r rfac value r sexp value r spacing value r sstiff value r str value r uaxial value r ushear value PLOT axial mat reinforce reinforce rshear PRINT property reinf reinforce

33 STRUCTURAL ELEMENTS Example Application Reinforced Slope The rock slope shown in Figure 1.6 is 40 m high, with foliation planes parallel to the slope face. The planes dip at an angle of 76,andhavea4mspacing. Two nearly horizontal joints intersect the slope face at a dip angle of 2.5. The friction angle of all joints is 6. The slope is not stable, and fails in a reverse-toppling mode, as shown in Figure 1.7. The slope is stabilized by adding two horizontal lines of local reinforcement. The mobilized axial forces in the reinforcement are shown in Figure 1.8. The input commands for this example are listed in Example 1.1: Example 1.1 Reinforced slope config cell round 0.5 edge 1 block 0,0 0,50 80,50 80,0 jregion id , , , ,10.0 jset angle spacing 4 origin 30,10 range jregion 1 crack (0,10) (32,10) crack (30,12.5) (65,14) crack (35,30) (70,31.5) delete range 0,30 10,50 group joint joint joint model area jks 1E8 jkn 1E8 jfriction 5.7 range group joint ; new contact default set jcondf joint model area jks=1e8 jkn=1e8 jfriction=5.7 change mat 1 property mat 1 density 2E3 fix range 0,80 10,20 set gravity= save reinf1.sav ; no support cycle 3000 save reinf2.sav ; add reinforcement reinforce , ,40.0 reinforce , ,20.0 property mat 1 r_astiff 1E8 r_length 1 r_sstiff 1E8 r_str 1E30 & r_uaxial 1E6 r_ushear 1E6 solve ratio 1.0E-5 save reinf3.sav

34 1-16 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE : Reinforced Slope (*10^1) UDEC (Version 5.00) LEGEND 11-Oct :17:22 cycle 0 time 0.000E+00 sec block plot Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 1.6 Slope with steeply dipping foliation planes JOB TITLE : Reinforced Slope (*10^1) UDEC (Version 5.00) LEGEND 11-Oct :16:15 cycle 3000 time 1.377E+01 sec block plot velocity vectors maximum = 1.346E E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 1.7 Slope failure by reverse toppling

35 STRUCTURAL ELEMENTS 1-17 JOB TITLE : Reinforced Slope (*10^1) UDEC (Version 5.00) LEGEND 11-Oct :20:59 cycle 1530 time 7.021E+00 sec block plot max axial force =-6.396E E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 1.8 Stabilization of slope by reinforcement

36 1-18 Special Features Structures/Fluid Flow/Thermal/Dynamics Global Shearing-Resistant Reinforcement (CABLE Command) In assessing the support provided by rock reinforcement, it is often necessary to consider not only the local restraint provided by reinforcement where it crosses discontinuities, but also the restraint to intact rock that may experience inelastic deformation in the failed region surrounding an excavation. Such situations arise in modeling inelastic deformations associated with failed rock and/or reinforcement systems (e.g., cable bolts) in which the bonding agent (grout) may fail in shear over some length of the reinforcement. Cable elements in UDEC allow the modeling of a shearing resistance along their length, as provided by the shear resistance (bond) between the grout and either the cable or the host medium. The cable is assumed to be divided into a number of segments of length L, with nodal points located at each segment end. The mass of each segment is lumped at the nodal points, as shown in Figure 1.9. Shearing resistance is represented by spring/slider connections between the structural nodes and the block zones in which the nodes are located. Deformable blocks must be specified to use the cable logic, and the blocks must be made deformable before the cable elements are installed. Each structural node is associated with a finite difference zone for calculation of shear forces between the cable and the zones. Reinforcing Element (steel) Grout Annulus Excavation m m Reinforcement Nodal Point Slider (cohesive strength of grout) m Axial Stiffness of Steel Shear Stiffness of Grout Figure 1.9 Conceptual mechanical representation of fully bonded reinforcement which accounts for shear behavior of the grout annulus

37 STRUCTURAL ELEMENTS Axial Behavior The axial behavior of conventional reinforcement systems may be assumed to be governed entirely by the reinforcing element itself. The reinforcing element is usually steel, and may be either a bar or cable. Because the reinforcing element is slender, it offers little bending resistance (particularly in the case of cables), and is treated as a one-dimensional member subject to uniaxial tension or compression. A one-dimensional constitutive model is adequate for describing the axial behavior of the reinforcing element. In the present formulation, the axial stiffness is described in terms of the reinforcement cross-sectional area, A, and the Young s modulus, E. The incremental axial force, F t, is calculated from the incremental axial displacement by F t = EA L ut (1.12) where u t = u i t i u [a] 1,u[b] 1 = u 1 t 1 + u 2 t 2 = (u [b] 1 u [a] 1 )t 1 + (u [b] 2 u [a] 2 )t 2, etc. are the displacements at the cable nodes associated with each cable element. Subscript 1 corresponds to the x-direction, and subscript 2 to the y-direction. The superscripts [a], [b] refer to the nodes. The direction cosines t 1,t 2 refer to the tangential (axial) direction of the cable segment. A tensile yield force limit (cb yield) and a compressive yield force limit (cb ycomp) can be assigned to the cable. Accordingly, cable forces that are greater than the tensile or compressive limits (Figure 1.10) cannot develop. If either cb yield or cb ycomp is not specified, the cable will have zero strength for loading in that direction. tensile force cb_yield compression 1 cb_ymod x asteel axial strain extension cb_ycomp compressive force Figure 1.10 Cable material behavior for cable elements

38 1-20 Special Features Structures/Fluid Flow/Thermal/Dynamics In evaluating the axial forces that develop in the reinforcement, displacements are computed at nodal points along the axis of the reinforcement, as shown in Figure 1.9. Out-of-balance forces at each nodal point are computed from axial forces in the reinforcement, as well as shear forces contributed through shear interaction along the grout annulus. Axial displacements are computed based on integration of the laws of motion using the computed out-of-balance axial force and a mass lumped at each nodal point Shear Behavior of Grout Annulus The shear behavior of the grout annulus is represented as a spring/slider system located at the nodal points shown in Figure 1.9. The shear behavior of the grout annulus during relative displacement between the reinforcing/grout interface and the grout/rock interface is described numerically by the grout shear stiffness cb kbond in (Figure 1.11). The maximum shear force that can be developed per length of element, Fs max /L, is limited by the cohesive strength of the grout (property keyword cb sbond). The limiting shear-force relation is depicted by the diagram in Figure Calculation of the relative displacement at the grout/rock interface uses an interpolation scheme to compute the displacement of the rock in the cable axial direction at the cable node. Each cable node is assumed to exist within an individual UDEC triangular zone (hereafter referred to as host zone). The interpolation scheme uses weighting factors that are based on the distance to each of the gridpoints of the host zone. The calculation of the weighting factors is based on satisfying moment equilibrium. force/length F s max L 1 cb_kbond relative shear displacement F s max L Figure 1.11 Grout material behavior for cable elements

39 STRUCTURAL ELEMENTS 1-21 For example, in computing the axial displacement of the grout/rock interface, the following interpolation scheme is used. Consider reinforcement passing through a constant-strain finite difference triangle making up part of the intact rock, as shown in Figure 1.12(a). The incremental x-component of displacement ( u xp ) at the nodal point is given by u xp = W 1 u x1 + W 2 u x2 + W 3 u x3 (1.13) where u x1, u x2, u x3 are the incremental gridpoint displacements; and W 1, W 2, W 3 are weighting factors. A similar expression is used for y-component displacements. The weighting factors W 1, W 2, W 3 are computed from the position of the nodal point within the triangle: W 1 = A 1 /A T (1.14) where A T is the total area of the finite-difference triangle; and A 1 is the area of the triangle in Figure 1.12(b). Incremental x- and y-displacements (Eq. (1.13)) are used at each calculation step to determine the new local reinforcing orientation. The axial component of displacement of the grout/rock interface is computed from the current orientation of the reinforcing segment. Forces generated at the grout/rock interface (F xp, F yp ) are distributed back to gridpoints according to the same weighting factors used previously: F x1 = W 1 F xp F x2 = W 2 F xp (1.15) F x3 = W 3 F xp where F x1, F x2 and F x3 are forces applied to the gridpoints.

40 1-22 Special Features Structures/Fluid Flow/Thermal/Dynamics Constant Strain Finite Difference Triangle 2 Gridpoint 1 3 Reinforcement Nodal Point (a) typical reinforcing element passing through a triangular sub-zone 2 A3 A1 1 A2 3 (b) areas used in determining weighting factors used to compute displacement of grout/medium interface Figure 1.12 Geometry of triangular finite difference zone and transgressing reinforcement used in distinct element formulation

41 STRUCTURAL ELEMENTS Normal Behavior at Grout Interface As explained above, an interpolated estimate of gridpoint velocity is made at each cable node. The velocity component normal to the average axial cable direction is transferred directly to the node (i.e., the cable node is slaved to the gridpoint motion in the normal direction). The node exerts no normal force on the grid if the cable segments on either side of the node are colinear. However, if the segments make an angle with each other, then a proportion of their axial forces will act in the mean normal direction. This net force acts on both the gridpoint and the cable node (in opposite directions). Thus, an initially straight cable can sustain normal loading if it undergoes finite deflection Cable Element Properties The cable elements used in UDEC require several input parameters: (1) cross-sectional area of cable; (2) mass density for cable; (3) elastic Young s modulus for cable; (4) tensile yield strength [force] of the cable; (5) compressive yield strength [force] of the cable; (6) extensional failure strain for the cable; (7) stiffness of the grout [force/cable length/displacement]; (8) cohesive capacity of the grout [force/cable length]; and (9) spacing. Note that property numbers are assigned to cable elements with the CABLE mats material property number for the cable material, and with the CABLE matg material property number for the grout. Each different cable can then be assigned geometric and material properties by specifying the PROPERTY command with the appropriate property keywords following the cable material property number and the grout material property number. For example, property mat=2 cb kbond = 1e9 cb sbond = 2e5 assigns a cable bond stiffness value of 10 9 and a cable bond strength value of to property number 2, which is defined (using the CABLE command) to be the property number for the grout. The area, density, modulus and yield-force resistance of the cable are usually readily available from handbooks, manufacturer s specifications, etc. A limiting extensional strain can also be defined for the cable; if not specified, the extensional strain is unlimited. The properties related to the grout are more difficult to estimate. The grout annulus is assumed to behave as an elastic, perfectly plastic solid. As a result of an incremental relative shear displacement,

42 1-24 Special Features Structures/Fluid Flow/Thermal/Dynamics u t, between the tendon surface and the borehole surface, the incremental shear force, F t, mobilized per length of cable is related to the grout stiffness, K bond : F t = K bond u t (1.16) K bond can be estimated from pull-out tests. Alternatively, the stiffness can be calculated from a numerical estimate for the elastic shear stress, τ G, obtained from an equation describing the shear stress at the grout/rock interface (St. John and Van Dillen 1983): τ G = G (D/2 + t) where G = grout shear modulus; D = reinforcing diameter; and t = annulus thickness. u ln(1 + 2t/D) (1.17) Consequently, the grout shear stiffness, K bond, is simply given by K bond = 2π G ln (1 + 2t/D) (1.18) In many cases, the following expression has been found to provide a reasonable estimate of K bond for use in UDEC: K bond 2π G 10 ln(1 + 2t/D) (1.19) The one-tenth factor helps to account for the relative shear displacement that occurs between the host-zone gridpoints and the borehole surface. This relative shear displacement is not accounted for in the present formulation. The maximum shear force per cable length in the grout is the bond cohesive strength. The value for bond cohesive strength can be estimated from the results of pull-out tests conducted at different confining pressures or, should such results not be available, the maximum force per length may be approximated from the peak shear strength (St. John and Van Dillen 1983): τ peak = τ I Q B (1.20) where τ I is approximately one-half of the uniaxial compressive strength of the weaker of the rock and grout, and Q B is the quality of the bond between the grout and rock (Q B = 1 for perfect bonding).

43 STRUCTURAL ELEMENTS 1-25 Neglecting frictional confinement effects, S bond may then be obtained from S bond = π(d + 2t) τ peak (1.21) Failure of reinforcing systems does not always occur at the grout/rock interface. Failure may occur at the reinforcing/grout interface, as is often true for cable reinforcing. In such cases, the shear stress should be evaluated at this interface. This means that the expressions (D + 2t) are replaced by (D)inEq. (1.21). The calculation of cable-element properties is demonstrated by the following example. A 25.4-mm (1 inch) diameter locked-coil cable was installed at 2.5-m spacing perpendicular to the plane of analysis. The reinforcing system is characterized by the properties cable diameter (D) hole diameter (D + 2t) cable modulus (E) cable ultimate tensile capacity grout compressive strength grout shear modulus (G) 25.4 mm 38mm 98.6 GPa MN 20 MPa 9 GPa Two independent methods are used in evaluating the maximum shear force in the grout. In the first method, the bond shear strength is assumed to be one-half the uniaxial compressive strength of the grout. If the grout-material compressive strength is 20 MPa and the grout is weaker than the surrounding rock, the grout shear strength is then 10 MPa. In the second method, reported pull-out data are used to estimate the grout shear strength. The report presents results for 15.9-mm (5/8 inch) diameter steel cables grouted with a 0.15-m (5.9 inch) bond length in holes of varying depths. The testing indicated capacities of roughly 70 kn. If a surface area of m 2 (0.15 m 0.05 m) is assumed for the cables, then the calculated maximum shear strength of the grout is 70 x 10 3 N m 2 = 9.33x106 N/m 2 = 9.33 MPa This value agrees closely with the 10 MPa estimated above, and either value could be used. Assuming that failure occurs at the cable/grout interface, the maximum bond force per length is (using Eq. (1.21), with D + 2t replaced by D) S bond = π ( m) (10 MPa) = 800 kn/m The bond stiffness, K bond, is estimated from Eq. (1.19). For the assumed values shown above, a bond stiffness of N/m m is calculated.

44 1-26 Special Features Structures/Fluid Flow/Thermal/Dynamics Values for K bond, S bond, E and tensile yield force are divided by 2.5 to account for the 2.5-m spacing of cables perpendicular to the modeled cross-section (see Section 1.5.1). The final input properties for UDEC are asteel m 2 cb kbond N/m/m cb sbond N/m cb ymod 40 GPa cb yield N Mass scaling is performed automatically to adjust the cable mass to achieve a timestep that coincides with that calculated for the UDEC model without cables. If the grout stiffness or axial modulus of the cable element is very high, it may be necessary to reduce the timestep (using the FRACTION command) to avoid numerical instability errors Pretensioning Cable Elements Cable elements may be pretensioned in UDEC by assigning the optional value preten with the CABLE command. A positive value for pretensioning assigns an axial force into the cable element(s) described by that CABLE command. Note that the cable with specified pretension is unlikely to be in equilibrium with other elements of the UDEC model to which it is initially linked. In other words, some displacements of the cable nodes and linked blocks are probably required to achieve equilibrium. These displacements will likely result in some loss of the initial pretension. In practice, pretensioned elements may be fully grouted, or they may be left ungrouted over part of their length. In either case, some form of anchorage is provided at the ends of the cable during pretensioning. This process can be simulated in UDEC by the application of several commands. First, the CABLE command is used to define the geometry for the cable, the property numbers for the anchored ends and the ungrouted section (the first-node and last-node keywords may be used to do this), and the pretension force. PROPERTY commands need to be used to define the different properties that exist at the anchored ends and the middle section. The ungrouted section is initially left free, so cb sbond should be 0.0 for that property. The model may be cycled at this point to allow the pretension force to distribute to the rock mass. The procedure for subsequent grouting of the free length is to simply change the cb sbond values for the free section to appropriate values for a grouted section.

45 STRUCTURAL ELEMENTS Estimating the Maximum Length for Cable Element Segments If the segment lengths for a cable element are too long, failure by grout slippage cannot occur because the cable material yield force will be reached before the shear resistance along the grout can be mobilized. It is recommended that at least two or three segments be provided along the development length of a cable in order to account for the possibility of grout slippage. The development length is the length of bonded reinforcing required to develop the axial capacity of the cable. The development length, l d, can be estimated from the grout bond strength, S bond, and the cable yield force capacity, F y, using the expression l d = F y S bond (1.22) Connecting Cable Elements to Beam Elements and to Other Cable Elements A cable element can be connected to a structural (beam) element, for example, to simulate a cable bolt connected to a tunnel lining. The connect keyword, specified at the end of a CABLE command, will position the cable node closest to a structural (beam) element node to coincide with the beam node. The structural elements must be defined first. The connection to the structural element node is not allowed to fail. The cable node connected to the structural element node will not appear in the list of the cable element nodes. The node ID number for the missing cable node will be the ID number of the structural element node to which the cable is connected. A new cable can be connected to the end of an existing cable by adding the keyword extend to the end of the CABLE command for the new cable. The end node of the new cable closest to the existing cable end node will be connected to the existing cable, and have the same node ID as the existing node Summary of Commands Associated with Cable Elements All of the commands associated with local reinforcement elements are listed in Table 1.2. See Section 1 in the Command Reference for a detailed explanation of these commands.

46 1-28 Special Features Structures/Fluid Flow/Thermal/Dynamics CABLE Table 1.2 Commands associated with cable elements x1 y1 x2 y2 npoint mats matg <preten> <keyword> <connect> <extend> <first node> <last node> CABLE delete <range> CHANGE cable matg <mats> PROPERTY mat mats keyword cb area cb density cb fstrain cb spacing cb thexp cb ycomp cb yield cb ymod PROPERTY mat matg keyword cb kbond cb sbond PLOT cable keyword afail axial element fail gfail node number sdisp shear strain svel xdisp xvel ydisp yvel mat cable PRINT property cable cable value value value value value value value value value value

47 STRUCTURAL ELEMENTS Example Application Pull-Test for a Grouted Cable Anchor The most common method for determination of cable bolt properties is to perform pull-out tests on small segments of grouted cables in the field. Typically, segments from 10 to 50 cm in length are grouted into boreholes. The ends of these segments are pulled with a jack mounted to the surface of the tunnel, and connected to the cable via a barrel-and-wedge type anchor. The force applied to the cable and the deformation of the cable are plotted to produce an axial force-deflection curve. From this curve, the peak shear strength of the grout bond is normally determined and converted to a strength in tons/m cable length. In this example we simulate a pull-test on a single 15.2-mm diameter cable. The cable material properties are cable area 181 mm 2 cable bond length 0.5 m cable modulus (E) 98.6 GPa cable ultimate tensile capacity MN We select two properties for the grout: grout bond stiffness grout cohesive strength N/m/m N/m These values are representative of a relatively weak grout (e.g., see Hyett et al. 1992) We apply a load to the cable by gluing a small block to the end of the cable; we can pull the cable by pulling the block. The cable is attached to the small block by assigning a high grout shear-stiffness and shear-strength to the cable node embedded in the small block. The grout properties are changed for the one cable node by changing the material number assigned to this node. The material number integer is changed by adding the keyword phrase first node 2 to the end of the CABLE command. This changes the material property number for the first node of this cable to 2. FISH function pullf is used to monitor the pull force on the cable (pull force per cable length). The pull force is determined from the sum of reaction forces that develop on the block as the cable is pulled. Example 1.2 presents the data file for this model: Example 1.2 Simulation of a pull-test for a grouted cable anchor round 1E-3 edge 2E-3 block 0,0 0,1 1,1 1,0 crack (0.5,0) (0.5,1) crack (0.4,0) (0.4,1) crack (0.4,0.4) (0.5,0.4)

48 1-30 Special Features Structures/Fluid Flow/Thermal/Dynamics crack (0.4,0.5) (0.5,0.5) delete range 0,0.4 0,1 delete range 0.4,0.5 0,0.4 delete range 0.4, ,1 gen quad 0.13,0.4 range 0.5,1 0,1 gen quad 0.11 range 0.4, ,0.5 group zone block zone model elastic density 2.5E3 bulk 5E9 shear 3E9 range group block group joint joint joint model area jks 1E11 jkn 1E11 jfriction 30 range group joint ; new contact default set jcondf joint model area jks=1e11 jkn=1e11 jfriction=30 save pulltest1.sav ; cable 0.48, , first_node 2 property mat 1 cb_area 1.81E-4 cb_density 7.5E3 cb_fstrain 1E30 cb_ycomp & 1E10 cb_yield 2.32E5 cb_ymod 9.86E10 cb_kbond 1.12E7 cb_sbond 1.75E5 & cb_spacing 1 property mat 2 cb_area 1.81E-4 cb_density 7.5E3 cb_fstrain 1E30 cb_ycomp & 1E10 cb_yield 2.32E5 cb_ymod 9.86E10 cb_kbond 1E10 cb_sbond 1E10 & cb_spacing 1 save pulltest2.sav ; ;Name:find_block def find_block ib_rock = b_near(.5,.5) ib_block = b_near(.7,.5) end find_block ;Name:pullf ;Input:x_loc/float/0.5/x location ;Input:x_tol/float/0.02/tolerance ; ; monitor axial force and cable displacement ; pullf : pull force per cable length ; x_disp : displacement of cable end def pullf sum = 0.0 x_plus = x_loc + x_tol ib = block_head loop while ib # 0 if ib = ib_rock then ig = b_gp(ib) loop while ig # 0 ibou=gp_bou(ig) ; index of boundary corner associated with gridpoint if (imem(ibou+2)) = 4 then

49 STRUCTURAL ELEMENTS 1-31 x_pos = gp_x(ig) if x_pos <= x_plus then sum = sum + fmem(ibou + 4) endif else sum = sum + gp_xforce(ig) endif ig = gp_next(ig) endloop endif ib = b_next(ib) endloop pullf = sum / 0.5 x_disp = * step * tdel end set x_loc=0.5 x_tol=0.02 pullf history pullf history x_disp history ncyc 50 save pulltest3.sav ; hide 0.4, ,0.5 boundary xvelocity 0 range 0.499, E-2,1.01 show boundary xvelocity -5E-3 range 0.39, ,0.52 save pulltest4.sav ; cycle save pulltest5.sav Figure 1.13 displays the axial force distribution in the cable at the end of the test. The total pull force versus displacement results from the test are shown in Figure As this figure shows, the peak load is very similar to the input value for grout cohesive strength. The loading slope also compares closely with the input value for grout bond stiffness. This is because the grout is much softer than the cable material. If the stiffness of the grout is increased, this slope will vary from the grout stiffness because the loading slope reflects the combined deflection due to both strain in the cable and the relative displacement of the grout. In general, the bond stiffness can be back-calculated by adjusting this stiffness to fit UDEC results to a pull-test force-displacement curve. The cable shear-bond strength will, in general, increase with increasing effective pressure acting on the cable. The pressure dependency is not accounted for directly in the present formulation. However, it is possible to add this dependency through the use of a FISH routine. Alternatively, rockbolt elements can be used to simulate this pressure dependency (see Section ).

50 1-32 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE : Cable Pull Test UDEC (Version 5.00) LEGEND Oct :07:34 cycle time 4.147E+00 sec block plot Axial Force on Structure Type # Max. Value cable E+04 displacement vectors maximum = 2.073E E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.13 Axial force and displacement vectors for pull-test JOB TITLE : Cable Pull Test UDEC (Version 5.00) (e+005) 1.80 LEGEND 11-Oct :07:34 cycle time 4.147E+00 sec history plot Y-axis: 1 - Fish: pullf X-axis: 2 - Fish: x_disp Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-001) Figure 1.14 Cable grout shear force versus displacement at node in small block

51 STRUCTURAL ELEMENTS Global Shearing- and Bending-Resistant Reinforcement (STRUCT rockbolt Command) The rockbolt element in UDEC is different from the cable bolt element in that it also provides bending-resistant behavior.* Rockbolts are two-dimensional elements with 3 degrees of freedom (two displacements and one rotation) at each end node. The formulation for the rockbolt element is identical to that for beams, as described in Section Rockbolt elements can yield in the axial direction, and can also simulate bolt breakage based upon a user-defined tensile failure strain limit. Rockbolts interact with UDEC via shear and normal coupling springs. The coupling springs are nonlinear connectors that transfer forces and motion between the rockbolt elements and the gridpoints associated with the block zone in which the rockbolt nodes are located. The formulation is similar to that for cable elements. The behavior of the shear coupling springs is identical to the representation for the shear behavior of grout, as described for cable elements in Section The behavior of the normal coupling springs includes the capability to model load reversal. The normal coupling springs are primarily intended to simulate the effect of the medium squeezing around the rockbolt. The formulations for the rockbolt segments and shear and normal coupling springs are described below Behavior of Rockbolt Segments A rockbolt element segment is treated as a linearly elastic material that may yield in the axial direction in both tension and compression. The behavior is identical to that prescribed for cable elements, as depicted in Figure Inelastic bending is simulated in rockbolts by specifying a limiting plastic moment. The present formulation in UDEC assumes that rockbolt elements behave elastically until they reach the plastic moment. This assumption is reasonably valid for rockbolt sections, because the difference between the moment necessary to produce the yield stress and the moment that results in yielding across the entire section is small. The section at which the plastic moment occurs can continue to deform without inducing additional resistance after it reaches this limit. The plastic-moment capacity sets the limit for the internal moments of structural-element segments for rockbolts. In addition, segments may break and separate at the nodes. Rockbolt breakage is simulated based upon a user-defined tensile failure strain limit (tfstrain). A strain measure, based upon adding the axial and bending plastic strains, is evaluated at each rockbolt node. The axial plastic strain, εpl ax,is accumulated based on the average strain of rockbolt element segments using the node. The bending plastic strain is averaged over the rockbolt, and then accumulated. The total plastic tensile strain, ε pl, is then calculated by * The rockbolt model was developed in collaboration with Geocontrol S.A., Madrid, Spain for application to analyses in which nonlinear effects of confinement, grout or resin bonding, or tensile rupture are important.

52 1-34 Special Features Structures/Fluid Flow/Thermal/Dynamics ε pl = εpl ax + d θ pl 2 L where d = rockbolt diameter; L = rockbolt segment length; and θ = average angular rotation over the rockbolt. (1.23) If this strain exceeds the limit tfstrain, the forces and moment in this rockbolt segment are set to zero, and the rockbolt is assumed to have failed Behavior of Shear Coupling Springs The shear behavior of the rockbolt/gridpoint interface is represented as a spring-slider system at the rockbolt nodal points. The system is similar to that illustrated for the cable/gridpoint interface in Figure 1.9. The shear behavior of the interface during relative displacement between the rockbolt nodes and the gridpoints is described numerically by the coupling spring shear stiffness (cs sstiff in Figure 1.33(b)): F s L = cs sstiff (u p u m ) (1.24) where F s = shear force that develops in the shear coupling spring (i.e., along the interface between the rockbolt element and the gridpoint); cs sstiff = coupling spring shear stiffness (cs sstiff); u p = axial displacement of the rockbolt; u m = axial displacement of the medium (soil or rock); and L = contributing element length. force/length cs_fric 1 cs_sstiff cs_scoh relative shear displacement perimeter a) Shear strength criterion b) Shear force versus displacement Figure 1.15 Material behavior of shear coupling spring for rockbolt elements

53 STRUCTURAL ELEMENTS 1-35 The maximum shear force that can be developed along the rockbolt/gridpoint interface is a function of the cohesive strength of the interface and the stress-dependent frictional resistance along the interface. The following relation is used to determine the maximum shear force per length of the rockbolt: Fs max L = cs scoh + σ c tan(cs sfric) perimeter (1.25) where cs scoh = cohesive strength of the shear coupling spring (cs scoh); σ c = mean effective confining stress normal to the rockbolt element; cs sfric = friction angle of the shear coupling spring (cs sfric); and perimeter = exposed perimeter of the element (perimeter). The effective confining stress acting on the rockbolt is based on the change in stress since installation. Stresses in the zone around the rockbolt are stored when the element is installed, and as calculation progresses, the effective confining stress around the element is calculated as the change in stress from the installation state. The mean effective confining stress normal to the element is defined by the equation σ c = ( σ nn + σ zz + p ) 2 (1.26) where p = pore pressure; σ zz = out-of-plane stress; and σ nn = σ xx n σ yy n σ xy n 1 n 2, n i = unit vectors relative to the local rockbolt segment axes. The limiting shear-force relation is depicted by the diagram in Figure 1.16(a). The input properties are shown in bold type on this figure. A user-defined table (cs cftable) can be specified to give a correction factor for the effective confining stress, in cases of non-isotropic stress, as a function of a deviatoric stress ratio. By default, the confining stress is given by Eq. (1.26). By specifying a table with cs cftable, factors are applied to the value of σ m to account for non-isotropic stresses. Softening as a function of shear displacement for the shear coupling-spring cohesion and friction angle properties can be prescribed via the user-defined tables cs sctable and cs sftable. The tables relate these properties to relative shear displacement. The same interpolation scheme as that employed for the cable elements is used to calculate the displacement of the block grid in the rockbolt axial direction at the rockbolt node.

54 1-36 Special Features Structures/Fluid Flow/Thermal/Dynamics Behavior of Normal Coupling Springs The normal behavior of the rockbolt/gridpoint interface is represented by a linear spring with a limiting normal force that is dependent on the direction of movement of the rockbolt node. The normal behavior during the relative normal displacement between the rockbolt nodes and the gridpoint is described numerically by the coupling spring normal stiffness (cs nstiff in Figure 1.16(b)): F n L = cs nstiff (up n un m ) (1.27) where F n = normal force that develops in the normal coupling spring (i.e., along the interface between the rockbolt element and the gridpoint); cs nstiff = coupling spring normal stiffness (cs nstiff); up n = displacement of the rockbolt normal to the axial direction of the rockbolt; u n m = displacement of the medium (soil or rock) normal to the axial direction of the rockbolt; and L = contributing element length. cs_nfric compressive force/length cs_ncoh cs_nten perimeter 1 cs_nstiff relative normal displacement cs_nfric tensile a) Normal strength criterion tensile force/length b) Normal force versus displacement Figure 1.16 Material behavior of normal coupling spring for rockbolt elements A limiting normal force can be prescribed to simulate the localized three-dimensional effect of the rockbolt pushing through the grid (e.g., a soil being squeezed around a single rockbolt). The limiting force is a function of a normal cohesive strength and a stress-dependent frictional resistance

55 STRUCTURAL ELEMENTS 1-37 between the rockbolt and the gridpoint. The following relation is used to determine the maximum normal force per length of the rockbolt: Fn max L = cs ncoh + σ c tan(cs nfric) perimeter (1.28) where cs ncoh = cohesive strength of the normal coupling spring (cs ncoh), which is dependent on the direction of loading; σ c = mean effective confining stress normal to the rockbolt element; cs nfric = friction angle of the normal coupling spring (cs nfric); and perimeter = exposed perimeter of the element (perimeter). The mean effective confining stress normal to the element is defined by Eq. (1.26) Rockbolt-Element Properties The rockbolt elements in UDEC require several input parameters (rockbolt property keywords are shown in parentheses): (1) cross-sectional area (area or radius) [length 2 ] of the rockbolt; (2) second moment of area (i) [length 4 ] (commonly referred to as the moment of inertia) of the rockbolt; (3) density (density) [mass/volume] of the rockbolt (optional used for dynamic analysis and gravity loading); (4) elastic modulus (e) [stress] of the rockbolt; (5) spacing (spacing) [length] (optional if not specified, rockbolts are considered to be continuous in the out-of-plane direction); (6) plastic moment (pmom) [force length] (optional if not specified, the moment capacity is assumed to be infinite); (7) tensile yield strength (yield) [force] of the rockbolt (if not specified, the tensile yield strength is zero); (8) compressive yield strength (ycomp) [force] of the rockbolt (if not specified, the compressive yield strength is zero); (9) tensile failure strain limit of the rockbolt (tfstrain); (10) exposed perimeter (perimeter) [length] of the rockbolt (i.e., the length of the rockbolt surface that is in contact with the medium); (11) stiffness of shear coupling spring (cs sstiff) [force/rockbolt length/displ.]; (12) cohesive strength of shear coupling spring (cs scoh) [force/rockbolt length];

56 1-38 Special Features Structures/Fluid Flow/Thermal/Dynamics (13) frictional resistance of the shear coupling spring (cs sfric) [degrees]; (14) number of table relating cohesion of shear coupling spring to relative shear displacement (cs sctable); (15) number of table relating friction angle of shear coupling spring to relative shear displacement (cs sftable); (16) number of table relating confining stress factor to deviatoric stress (cs cftable); (17) stiffness of normal coupling spring (cs nstiff) [force/rockbolt length/displ.]; (18) cohesive (and tensile) strength of normal coupling spring (cs ncoh) [force/rockbolt length]; (19) frictional resistance of the normal coupling spring (cs nfric) [degrees]; and (20) thermal expansion coefficient (thexp) (optional used for thermal analysis). The radius of the rockbolt element cross-section can also be prescribed instead of the area and moment of inertia. The area and moment of inertia will then be calculated automatically. The exposed perimeter of a rockbolt element and the properties of the coupling springs should be chosen to represent the behavior of the rockbolt/medium interface commensurate with the problem being analyzed. The rockbolt/rock interaction can be expressed in terms of a shear response along the length of the bolt as a result of axial loading and/or in terms of a normal response when the direction of loading is perpendicular to the rockbolt axis Commands Associated with Rockbolt Elements All of the commands associated with rockbolt elements are listed in Table 1.3. This includes the commands associated with the generation of rockbolts, and those required to plot and print rockboltelement variables. See Section 1.3 in the Command Reference for a detailed explanation of these commands.

57 STRUCTURAL ELEMENTS 1-39 Table 1.3 STRUCT Commands associated with rockbolt elements keyword rockbolt keyword begin end prop segment keyword node xy keyword node xy np ns delete <n1 n2> node n x y node n* keyword fix free initial load mat pin slave unslave n n <x><y><r> <x><y><r> keyword xdis xvel ydis yvel rvel fx fy m <x><y> m <x><y> * For the keywords fix, free, initial, load and pin, a range of nodes can be specified with the phrase range n1 n2. value value value value value

58 1-40 Special Features Structures/Fluid Flow/Thermal/Dynamics Table 1.3 Commands associated with rockbolt elements (continued) STRUCT prop np keyword area value cs ncoh value cs nfric value cs nstiff value cs scoh value cs sfric value cs sstiff value cs sctable n cs sftable n cs cftable n density value e value i value perimeter value pmom value radius value spacing value tfstrain value thexp value ycomp value yield value

59 STRUCTURAL ELEMENTS 1-41 Table 1.3 Commands associated with rockbolt elements (continued) PLOT rockbolt keyword afail avel <n> axial <n> ifail inormal <n> interface ishear <n> moment <n> number sdisp shear <n> svel xdisp <n> xvel <n> ydisp <n> yvel <n> mat rockbolt PRINT rockbolt keyword element node

60 1-42 Special Features Structures/Fluid Flow/Thermal/Dynamics Example Application Rockbolt Pullout Tests The most common method for determination of rockbolt properties is to perform pullout tests on small segments of rockbolts in the field. Typically, segments of 50 cm in length, or longer, are grouted into boreholes. The ends of these segments are pulled with a jack mounted to the surface of the tunnel, and connected to the rockbolt via a barrel-and-wedge type anchor. The force applied to the rockbolt and the deformation of the rockbolt are plotted to produce an axial force-deflection curve. From this curve, the peak shear strength of the grout bond is determined. The results of simulated pullout on one-half meter segments are illustrated in this example. The data file in Example 1.3 contains several variations of a single rockbolt pull-test. The rockbolt end node is pulled at a small, constant y-direction velocity, as indicated in Figure A FISH function ff is used to sum the reaction forces and monitor nodal displacement generated during the pull-tests. JOB TITLE : Rockbolt Pull Test UDEC (Version 5.00) LEGEND 11-Oct :49:08 cycle 0 time 0.000E+00 sec block plot RockBolt elements plotted Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.17 Rockbolt element in grid; velocity applied at top end node

61 STRUCTURAL ELEMENTS 1-43 Example 1.3 Rockbolt pullout tests round 0.01 edge 0.02 block 0,0 0, , ,0 gen quad 0.11 group zone block zone model elastic density 2E3 bulk 5E9 shear 3E9 range group block boundary yvelocity 0 range -0.01, ,0.61 struct rockbolt begin 0.2,0.1 end 0.2,0.7 seg 12 prop 1 struct prop 1 e 200e9 a 5e-4 cs_scoh 1.0e5 cs_sstiff 2.0e7 per 0.08 struct prop 1 yield 2.25e5 i 2e-8 dens struct node 13 fix y struct node 13 ini yvel 8e-2 save p1.sav ; ; --- Fish functions --- ; ff : Pull force in bolt ; dd : Displacement of rockbolt end ca boucnr.fin ca str.fin def _find_end_node _inode = str_node_head _end_node = 0 loop while _end_node = 0 _yp = fmem(_inode+$sndy) ; node 13 if _yp > 0.69 then _end_node = _inode endif _inode = imem(_inode+$sndnext) end_loop end _find_end_node def ff1 whilestepping ; node 13 nadd = _end_node dd = fmem(nadd+$sndu2) ffnode = fmem(nadd+$kndfor2) ffbou = 0.0 loop jj (1,5) xx = (jj-1) * 0.1 ig1 = gp_near(xx,0.6) ibou1 = gp_bou(ig1)

62 1-44 Special Features Structures/Fluid Flow/Thermal/Dynamics fb1 = fmem(ibou1+$kbdfy) ffbou = ffbou+fb1 endloop end hist dd hist ffbou hist ffnode hist ydis save p2.sav ; ; --- pull out tests - single 25 mm rockbolt (20 mm deformation) ; ; --- default behavior --- restore p2.sav history unbalanced cycle save p3.sav ; ; --- cohesion softening --- restore p2.sav stru pro 1 cs_sctable=100 table e e4 ;change in cohesion with relative shear displ. step save p4.sav ; ; --- confinement = 5 MPa (in-plane) --- restore p2.sav stru pro 1 e 200e9 a 5e-4 cs_sstiff 2.00e7 per 0.08 stru pro 1 yield 2.25e5 ; ult. tens. str. (450 MPa)*area = Force stru pro 1 i=2e-8 ; 0.25*pi*rˆ4 stru pro 1 cs_sfric=45 stru pro 1 cs_scoh=0.0 step 1 insitu str -5e6 0 0 szz 0 bou xv 0 range xr bou xv 0 range xr step save p5.sav ; ; --- confinement = 5 MPa (in-plane) with conf. str. table --- restore p2.sav stru pro 1 e 200e9 a 5e-4 cs_sstiff 2.00e7 per 0.08 stru pro 1 yield 2.25e5 ;ult. tens. strength (450 MPa) * area = Force stru pro 1 i=2e-8 ; 0.25*pi*rˆ4 stru pro 1 cs_sfric=45 stru pro 1 cs_scoh=0.0

63 STRUCTURAL ELEMENTS 1-45 ; define table for confining stress correction factor table struct prop 1 cs_cftable 1 ; note : (snn-szz)/(snn+szz) is 1 : cfac = 0.36 step 1 insitu str -5e6 0 0 szz 0 bou xv 0 range xr bou xv 0 range xr step save p6.sav ; ; --- tensile rupture --- restore p2.sav stru pro 1 e 200e9 a 5e-4 cs_scoh 1.00e5 cs_sstiff 2.00e7 per 0.08 stru pro 1 yield 1.0e5 ; ult. tens. strength (200 MPa) * area = Force stru pro 1 i=2e-8 ; 0.25*pi*rˆ4 stru pro 1 cs_sfric=45 stru pro 1 tfs = 5e-2 step 1 insitu str -5e6 0 0 szz 0 bou xv 0 range xr bou xv 0 range xr step save p7.sav ret

64 1-46 Special Features Structures/Fluid Flow/Thermal/Dynamics In the first test, confining stress dependence on the rockbolt shear bond strength is neglected. The resulting axial force-deflection plot is shown in Figure The peak force is approximately 50 kn. JOB TITLE : Rockbolt Pull Test UDEC (Version 5.00) (e+004) 5.00 LEGEND 11-Oct :50:00 cycle time 2.357E-01 sec history plot Y-axis: 2 - Fish: ffbou X-axis: 1 - Fish: dd Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-002) Figure 1.18 Rockbolt pull force (N) versus rockbolt axial displacement (meters) for a single 25 mm grouted rockbolt

65 STRUCTURAL ELEMENTS 1-47 In the second test, displacement weakening of the shear bond strength is introduced using the cs sctable property. The displacement weakening relation to shear displacement is defined in table 100. The results are shown in Figure 1.19: JOB TITLE : Rockbolt Pull Test UDEC (Version 5.00) (e+004) 3.50 LEGEND 11-Oct :53:38 cycle time 2.357E-01 sec history plot Y-axis: 2 - Fish: ffbou X-axis: 1 - Fish: dd Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-002) Figure 1.19 Rockbolt pull force (N) versus rockbolt axial displacement (meters) for a single 25 mm grouted rockbolt with displacement weakening

66 1-48 Special Features Structures/Fluid Flow/Thermal/Dynamics The rockbolt shear bond strength will, in general, increase with increasing effective pressure acting on the rockbolt. A linear law is implemented in UDEC, whereby the rockbolt shear strength is defined as a constant (cs scohesion) plus the effective pressure on the rockbolt multiplied by the rockbolt perimeter (perimeter) times the tangent of the friction angle (cs sfriction). The pressure dependence is activated automatically by issuing the rockbolt properties perimeter and cs sfriction. In the third test, a 5 MPa in-plane confining stress is applied after the rockbolt is installed. Note that one calculational step is taken, in order to assign the rockbolt properties before the confining stress is applied. The results are shown in Figure 1.20: JOB TITLE : Rockbolt Pull Test UDEC (Version 5.00) (e+005) 1.00 LEGEND 11-Oct :36:11 cycle time 2.357E-01 sec history plot Y-axis: 2 - Fish: ffbou X-axis: 1 - Fish: dd Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-002) Figure 1.20 Rockbolt pull force (N) versus rockbolt axial displacement (meters) for a single 25 mm grouted rockbolt with 5 MPa in-plane confinement and zero out-of-plane confinement

67 STRUCTURAL ELEMENTS 1-49 In the fourth test, the property cs cftable is used to define the confining stress applied to the rockbolt, accounting for the reduced affect of the out-of-plane stress and the in-plane stress normal to the bolt. Table 1 is used to apply the reduction factor. The results are shown in Figure Note that the pullout resistance is reduced compared to the previous case (compare Figure 1.21 to Figure 1.20). JOB TITLE : Rockbolt Pull Test UDEC (Version 5.00) (e+005) 1.00 LEGEND 11-Oct :36:43 cycle time 2.357E-01 sec history plot Y-axis: 2 - Fish: ffbou X-axis: 1 - Fish: dd Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-002) Figure 1.21 Rockbolt pull force (N) versus rockbolt axial displacement (meters) for a single 25 mm grouted rockbolt with 5 MPa in-plane confinement plus a reduction factor and zero out-of-plane confinement

68 1-50 Special Features Structures/Fluid Flow/Thermal/Dynamics In the fifth test, yield is used to define the limiting axial yield force (100 kn) of the bolt, and tfstrain is used to define the plastic strain (0.05) at which the bolt ruptures. The results are shown in Figure 1.22: JOB TITLE : Rockbolt Pull Test UDEC (Version 5.00) (e+005) 1.20 LEGEND 11-Oct :42:14 cycle time 2.004E-01 sec history plot Y-axis: 2 - Fish: ffbou X-axis: 1 - Fish: dd Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-002) Figure 1.22 Rockbolt pull force (N) versus rockbolt axial displacement (meters) for a single 25 mm grouted rockbolt with tensile rupture

69 STRUCTURAL ELEMENTS Example Application Rockbolt Shear Tests Two shear tests are performed in this example. The tests use the same model as the pullout tests. In this case, though, a horizontal velocity is applied to the top rockbolt node. The data file is listed in Example 1.4. Note that normal coupling spring properties are now included. Example 1.4 Rockbolt shear tests round 0.01 edge 0.02 block 0,0 0, , ,0 gen quad 0.12 group zone block zone model elastic density 2E3 bulk 5E9 shear 3E9 range group block boundary xvelocity 0 range -0.01, ,0.01 boundary yvelocity 0 range -0.01, ,0.01 boundary xvelocity 0 range -0.01, ,0.61 boundary xvelocity 0 range 0.29, ,0.61 struct rockbolt begin 0.15,0.1 end 0.15,0.625 seg 25 prop 4 stru prop 4 e 200e9 a 5e-4 cs_scoh 1.00e5 cs_sstiff 2.00e7 per 0.08 stru prop 4 yield 2.25e5 ; ult. tens. strength (450 MPa) * area = Force stru prop 4 i=2e-8 ; 0.25*pi*r 4 stru prop 4 cs_nstiff 1e10 cs_ncoh 2e6 cs_nfric=45 stru prop 4 dens ; stru node 26 fix x stru node 26 ini xvel 8e-2 save shear1.sav ; call boucnr.fin call str.fin ; --- Fish functions --- ; ff : Pull force in bolt ; dd : Displacement of rockbolt end def _find_node _inode = str_node_head _this_node = 0 loop while _this_node = 0 _yp = fmem(_inode+$sndy) _ydis = abs(_yp - _ynode) if _ydis <.01 then _this_node = _inode endif _last_node = _inode _inode = imem(_inode+$sndnext) if _inode = 0 then

70 1-52 Special Features Structures/Fluid Flow/Thermal/Dynamics _this_node = _last_node endif end_loop _find_node = _this_node end def _nodes _ynode = _node26 = _find_node _ynode = _node25 = _find_node _ynode = _node24 = _find_node end _nodes def ff1 whilestepping ; node 26 nadd = _node26 dd = fmem(nadd+$sndu1) ffnode = fmem(nadd+$kndfor1) dx26 = fmem(nadd+$sndu1) dy26 = fmem(nadd+$sndu2) dr26 = fmem(nadd+$sndur) fx26 = fmem(nadd+$sndf1) ; node 25 nadd = _node25 dx25 = fmem(nadd+$sndu1) dy25 = fmem(nadd+$sndu2) dr25 = fmem(nadd+$sndur) fx25 = fmem(nadd+$sndf1) ; node 24 nadd = _node24 dx24 = fmem(nadd+$sndu1) dy24 = fmem(nadd+$sndu2) dr24 = fmem(nadd+$sndur) fx24 = fmem(nadd+$sndf1) ; ffbou = 0.0 loop jj (1,7) yy = (jj-1) * 0.1 ig1 = gp_near(0.0,yy) ibou1 = gp_bou(ig1) fb1 = fmem(ibou1+$kbdfx) ffbou = ffbou+fb1 endloop loop jj (1,7)

71 STRUCTURAL ELEMENTS 1-53 yy = (jj-1) * 0.1 ig1 = gp_near(0.3,yy) ibou1 = gp_bou(ig1) fb1 = fmem(ibou1+$kbdfx) ffbou = ffbou+fb1 endloop loop jj (1,2) xx = (jj) * 0.1 ig1 = gp_near(xx,0.0) ibou1 = gp_bou(ig1) fb1 = fmem(ibou1+$kbdfx) ffbou = ffbou+fb1 endloop end ; --- Histories --- hist n 100 hist dd hist ffbou hist ffnode hist unbal hist ydis hist dx26 dy26 dr26 fx26 hist dx25 dy25 dr25 fx25 hist dx24 dy24 dr24 fx24 save shear2.sav ; --- shear test --- cycle save shear3.sav ; --- bolt ruptures restore shear2.sav stru prop 4 pmom=5e3 tfs=1e-2 hist ncyc 70 step save shear4.sav ret Figure 1.23 shows the plot of shear force versus shear displacement for a non-yielding bolt. Figure 1.24 shows the rockbolt geometry at the end of the test. The large displacement of the rockbolt near the rock surface is a result of the failure of the normal coupling springs, which simulates the crushing of the rock. In the second test, pmom is specified to define a limiting moment (5000 N-m) of the bolt, and tfstrain is set to define a limiting plastic strain (0.01) at which the bolt ruptures. The results are shown in Figures 1.25 and 1.26.

72 1-54 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE : Rockbolt Shear Test UDEC (Version 5.00) (e+005) 1.40 LEGEND 11-Oct :24:46 cycle time 3.527E-01 sec history plot Y-axis: 2 - Fish: ffbou X-axis: 1 - Fish: dd Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-002) Figure 1.23 Rockbolt shear force (N) versus rockbolt shear displacement (meters) for a single 25 mm grouted rockbolt JOB TITLE : Rockbolt Shear Test UDEC (Version 5.00) LEGEND Oct :24:46 cycle time 3.527E-01 sec zones in fdef blocks RockBolt elements plotted Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.24 Deformed shape of 25 mm diameter rockbolt at end of shear test

73 STRUCTURAL ELEMENTS 1-55 JOB TITLE : Rockbolt Shear Test UDEC (Version 5.00) (e+005) 1.00 LEGEND 11-Oct :31:11 cycle time 3.527E-01 sec history plot Y-axis: 2 - Fish: ffbou X-axis: 1 - Fish: dd Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-002) Figure 1.25 Rockbolt shear force (N) versus rockbolt shear displacement (meters) for a single 25 mm grouted rockbolt with tensile rupture JOB TITLE : Rockbolt Shear Test UDEC (Version 5.00) LEGEND Oct :31:11 cycle time 3.527E-01 sec zones in fdef blocks RockBolt elements plotted Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.26 Deformed shape of 25 mm diameter rockbolt following rupture at end of shear test

74 1-56 Special Features Structures/Fluid Flow/Thermal/Dynamics 1.3 Surface Support Two types of structural elements are available in UDEC to simulate support placed on exposed rock surfaces. The first type consists of two-dimensional structural (beam) elements. Surface linings for tunnels and exposed slopes are typically thin, and their characteristic response to bending usually cannot be neglected. The beam element formulation provides an effective method to include bending effects. The beam elements are attached to a rock surface via spring connections oriented both radially and tangentially with respect to the support structure. The second type of structural element consists of one-dimensional support members that are attached to two boundaries of an interior surface. The support member has no independent degrees of freedom, but simply imposes forces on the surfaces to which it is connected. Support members are intended to model props or packs that are placed to support an underground opening. The characteristic behaviors of these two support types are described in the following sections Structural Beam Elements (STRUCT generate Command) The structural element method is well-documented in structural engineering texts. The use of beam elements in two-dimensional linear analysis of excavation support is reported by Dixon (1971), Brierley (1975) and Monsees (1977), among others. Paul et al. (1983) present analyses using beam elements that include nonlinear behavior. Analysis of any support structure is initiated by discretization of the structure into a number of elements whose response to axial, transverse and flexural loads can be represented in matrix form, as shown in Figure The rock-structure interface is represented by springs connected between the structural element nodes and the UDEC model. The structural beam elements can be connected to either rigid blocks or deformable blocks Structural (Beam) Element Formulation Either an implicit or explicit formulation may be used in analyzing the behavior of a support structure composed of beam elements and interface stiffnesses. In the first formulation (implicit), a global stiffness matrix is formed for the entire structure. The size of the stiffness matrix is reduced by deleting free nodes (i.e., those nodes that are not located at the rock-support interface). This is possible because these nodes are not subjected to directly imposed external loads or displacements by the surrounding medium. The resultant efficiency, however, limits straightforward application of this formulation to quasi-static problems involving linear elastic behavior. This formulation does not provide information about failure mechanisms or ultimate capacities of interior supports. However, factors of safety based on lining stresses should be conservative since they do not take into account the highly indeterminate nature of a lining in contact with the rock. A detailed description of this formulation, its use with distinct elements, and numerous other examples are presented by Lorig (1984).

75 STRUCTURAL ELEMENTS 1-57 L v, S 2 2 u, T 2 2 m 2,M v, S m u, T,M 1 1 E,I,A Structural Element Sign Convention T 1 A u 1 S 1 0 SYM. 12 I v L 2 1 M 1 0 = E L 6 I L 4 I 1 T 2 -A 0 0 A u 2 S I L 2 6 I - 0 L 12 I v L 2 2 M I - 2 I 0 L 6 I - 4 I 2 L Figure 1.27 Local stiffness matrix for structural element representation of excavation support In the second formulation (explicit), local stiffness matrices are used following division of the structure into segments with the distributed mass of the structure lumped at nodal points, as shown in Figure Forces generated in support elements are applied to the lumped masses, which move in response to unbalanced forces and moments in accordance with the equations of motion. This formulation has two desirable characteristics: (1) slip between support and excavation periphery is modeled in a manner identical to block interaction along a discontinuity; and (2) large displacements with nonlinear material behavior are readily accommodated. These capabilities are illustrated in Figure 1.29, in which a roof block loads and displaces a hypothetical four-element interior structural support.

76 1-58 Special Features Structures/Fluid Flow/Thermal/Dynamics Excavation Periphery m 1. Lumped Mass m 2. Structural Element 3. Interface Lining Interior m Figure 1.28 Lumped mass representation of structure used in explicit formulation Figure 1.29 Demonstration of interface slip and large displacement capabilities of explicit structural element formulation

77 STRUCTURAL ELEMENTS 1-59 The structural (beam) elements include an elastic-plastic material model that incorporates bending resistance, limiting bending moments and yield strengths of the beam material.* The material model can simulate inelastic behavior that is representative of common surface-lining materials. This includes materials (such as steel) that behave in a ductile manner, as well as non-reinforced and reinforced cementatious materials (such as concrete and shotcrete) that can exhibit either brittle or ductile behavior. Note that shear failure is not included in the material model. The behavior of the material model can be shown on a moment-thrust interaction diagram, such as that given in Figure Moment-thrust diagrams are commonly used in the design of concrete columns. These diagrams illustrate the maximum force that can be applied to a typical section for various eccentricities. The ultimate failure envelopes for non-reinforced and reinforced cementatious materials are similar. However, reinforced materials have a residual capacity that remains after failure at the ultimate load. Non-reinforced cementatious materials typically have no residual capacity. Compressive Force, P 1 Sc=Fc Compression Failure Sc= Fc St<Ft ultimate failure envelope Sc= Fc Fc: Compression Strength Ft : Tension Strength Sc : Compression Stress St : Tension Stress Sc = P/A + Mc/I St = P/A - Mc/I M = Pe P e failure envelope for cracked section 3 St=Ft Sc<Fc balanced point e 1 St=Ft Bending Moment, M Tension Failure 2 Tensile Force, P Figure 1.30 St=Ft Typical moment-thrust diagram Interaction diagrams can be constructed by specifying the section geometry and compressive and tensile strengths (in terms of stress) for the material. The thickness and compressive and tensile strengths are input (with property keywords st ycomp and st yield, respectively), and the model uses this information to determine the ultimate capacity for various eccentricities (e in Figure 1.30). As * The inelastic material behavior model was developed with funding from the Norwegian Geotechnical Institute, NGI, Oslo, Norway, for application to the analysis of fiber-reinforced shotcrete.

78 1-60 Special Features Structures/Fluid Flow/Thermal/Dynamics the calculation progresses, the axial forces and moments in the structural elements are compared to the ultimate capacity. When a node reaches the ultimate capacity, a fracture flag is set, indicating that all future evaluations for that node will use the cracked failure envelope and the residual strength capacity (specified with the property keyword st scresid for residual compressive strength and st yresid for residual tensile strength). The diagram in Figure 1.30 is defined by the three points (1, 2 and 3) shown in the figure. The axial force, P, and moment, M, at these three points are calculated: at Point 1 P 1 = F c A M 1 = 0 (1.29) at Point 2 P 2 = F t A M 2 = 0 (1.30) at Point 3 P 3 = (F c + F t ) A 2 M 3 = I (F c F t ) h (1.31) where F c and F t are the initial compressive and tensile strengths of the material, A is the crosssectional area, I is the moment of inertial and h is the thickness of the liner section. The diagram in Figure 1.30 is developed assuming that the liner is initially uncracked. For unreinforced concrete or shotcrete, some cracking in the liner section may be permissible. The effect of cracking is approximated in the liner model by setting F t = 0 and introducing an extra point, Point 4, which extends the ultimate failure envelope when a tensile crack exists. The crack depth, h c,is generally limited to half the total section thickness, and can be related to the eccentricity, e 4, when the compressive stress reaches the compressive strength, F c, by the equation e 4 = h 6 + h c 3 (1.32) The axial force and moment at the extra point (Point 4) are then calculated as P 4 = 1 2 F ca(1 h c h ) M 4 = 1 12 F cah(1 + h c h 2(h c h )2 ) (1.33)

79 STRUCTURAL ELEMENTS 1-61 The crack depth ratio, h c /h, is assigned to beam elements as a property via the rcrack keyword. A parabolic shape of the failure envelope is often defined for the moment-thrust diagram in unreinforced liner design. The bending diagram can be input directly as a look-up table in UDEC. The table stores the P-M diagram, and then is assigned as the failure envelope by specifying the table number using the pmtable property keyword. An example application given in Section illustrates the inelastic material behavior of the structural (beam) element, and the construction of moment-thrust diagrams for the different cases of beam material behavior described above Structural Element Generation Structural elements are generated as a liner along a surface that is a boundary of a domain. The domain can be either an internal region in the model or the exterior (outer domain) boundary. Two methods are provided in UDEC to generate structural element liners. Liner Generated by Specifying Liner/Block Interface Contact Points In this method, a liner is created by defining a starting point and ending point of the interface contact points for the liner along the block internal or external boundary using the command struct gen begin x b,y b end x e,y e max value min value mat value The starting and ending point coordinates (x b,y b ) and (x e,y e ) are positioned, by default, at midpoints between block boundary corners (for rigid blocks) or at midpoints between boundary gridpoints (for deformable blocks). Additional structural nodes can be added along the liner surface by using the max keyword, which sets the maximum size for structural element segments along the liner surface. The keyword min can be used to prevent structural nodes along the liner surface from being too close to one another. By default, min is set to the minimum block edge length. Alternatively, all of the interface contact points can be generated individually by using the STRUCT beam table nt command to generate the liner. In this case, a table is first created using the TABLE command in which the x,y pairs of the table correspond to the interface contact points for the liner/block. Points in the table should lie within a tolerance of the rounding length to the block edge, in order for these points to be recognized as interface contact points for the liner. The liner can be generated one segment at a time by using the STRUCT beam begin x b,y b end x e,y e command to generate each beam segment along the liner surface. Also, structural nodes can be generated individually by using the STRUCT node command, and then beam segments can be created by connecting nodes with the command STRUCT beam begin node n1 end node n2. Liner Generated by Spraying In the second method, interface contact points for the liner/block interaction are sprayed from a central point located within the domain. (The central point must be within the domain that contains the surface to receive the lining.) The extent of the lined region is controlled by specifying (1) the angle of the first contact point (a positive angle is measured counterclockwise about the central point from the positive x-axis), and (2) the total angle of the surface to receive the lining (a positive angle is measured counterclockwise about the central point

80 1-62 Special Features Structures/Fluid Flow/Thermal/Dynamics from the angle of the first contact point). See Figure 1.31 for an illustration of these locations. If these limits are not specified, the entire boundary surface for the domain will be lined. An average number of contact points for the lining is selected by the user. Contact points will be generated to support all block edges along the surface. The total number of contact points generated can be greater than the average number specified. structural elements theta ( xc, yc) fang x Figure 1.31 Parameters to define structural element locations Layering Structural Element Segments Multiple sections of lining can be created along the same surface. For example, two different linings can be applied in layers to the same surface. Interaction will occur at the interface between the linings. There is no restriction to the number of layers that can be applied. Structural element contacts will be created between the layers. Different material properties can be assigned to the contacts between two layered segments by using the smat keyword with the STRUCT gen command to specify a separate material number for properties between two layers. The user must check for proper placement of structural element layers to be sure that they will interact. There must be at least one structural node of the new layer overlapping the previous layer for the layers to interact. Overlapping layers are detected if the new layer is within a space that is 0.4 times the thickness of the previous layer. The overlap space can be controlled with the stol keyword. The plotting command PLOT struct thick is helpful in determining whether layers are interacting. Connecting Structural Element Segments Structural nodes of different segments that are located at the same position can be connected by specifying the connect keyword at the end of the STRUCT gen command. The structural element segments can only connect at end nodes. The ends must be located within the rounding length tolerance, to connect. The result of connecting two structural element segments is one segment. There will be a skip in the node ID numbers because one node will be deleted at the connection. Cable nodes can also be connected to structural element nodes. See Section

81 STRUCTURAL ELEMENTS 1-63 Deleting Structural Elements Structural element segments can be deleted at any time in the calculation process by specifying the command STRUCT delete followed by a range that includes the centroid of the element segment. For example, to delete structural elements along a circular excavation boundary, use the command struct delete range annulus (0,0) Structural Element Properties The structural beam elements used in UDEC require three sets of input parameters: geometry parameters, constitutive model and material number parameters, and liner/block interface and structural element properties. These parameters are listed below. Geometry Parameters Geometry parameters generate the geometry and specify the general conditions of the structural element lining. This set of parameters can be given in two different forms. If the lining is generated by specifying individual liner/block contact points along a boundary surface, the input parameters are (1) x- and y-coordinates of starting and ending points along the lining surface; (2) maximum length of a structural element segment; (3) minimum length of a structural element segment; and (4) material number for element and interface properties. If the lining is generated by spraying the liner/block contact points within a domain, the parameters are (1) x- and y-coordinates of a point within the domain that contains the lined surface; (2) average number of interface contact points; (3) material number for element and interface properties; (4) angle of first contact point (positive angle is measured counterclockwise from the positive x-axis); (5) total angle of structural element lining (positive angle is measured counterclockwise from the first contact point); and (6) minimum length of a structural element segment.

82 1-64 Special Features Structures/Fluid Flow/Thermal/Dynamics Constitutive Model and Material Number Parameters The second set of parameters specifies (or changes) the element or liner/block interface material conditions. These parameters can be used to assign different conditions and properties to different portions of the lining. The parameters include (1) interface constitutive model; (2) material number for element properties; and (3) material number for interface contact properties. Liner/Block Interface and Structural Element Properties The third set specifies the material properties for the element and the interface. These properties are (1) elastic modulus [stress]; (2) Poisson s ratio; (3) density; (4) tensile yield strength [stress]; (5) residual tensile yield strength [stress]; (6) compressive yield strength [stress]; (7) residual compressive yield strength (stress); (8) ratio of crack depth to initial cross-section thickness; (9) P-M diagram look-up table; (10) interface normal stiffness [stress/unit displacement]; (11) interface shear stiffness [stress/unit displacement]; (12) interface cohesion [stress]; (13) interface friction [degrees]; (14) interface dilation [degrees]; (15) interface tensile strength [stress]; (16) element shape; (17) element second moment of inertia; (18) element thickness; (19) element cross-sectional area; (20) element width; and

83 STRUCTURAL ELEMENTS 1-65 (21) element spacing. Note that property numbers are assigned to structural elements and interfaces with the STRUCT gen mat command. Property numbers can be changed locally for structural element material, and for interface material, by using the STRUCT change mat n material property number for the structural element material, and the STRUCT change jmat n material property number for the interface material. Each different structural element and interface point can then be assigned geometric and material properties by specifying the PROPERTY command with the appropriate property keywords following the structural-element material property number and the interface material property number. For example, struct change jmat 2 property mat=2 if kn = 1e9 if coh = 2e5 assigns an interface stiffness value of 10 9, and an interface, cohesive strength value of to property number 2. Note that the interface contacts behave according to the jcons 5 constitutive model (i.e., the properties are specified in terms of stress and displacement). If the structural element has unit dimension in the out-of-plane direction, then only the thickness is required; the area, moment of inertia and shape factor are calculated automatically. For rectangular shapes, the shape factor is 5/6. Figure 1.32 lists shape factors and inertial moments for various shapes: b shape factor 5 6 inertial moment ab 3 12 a r 9 10 πr 4 4 r e ri e i π(r - r ) 4 web area total area Figure 1.32 Shape factors and inertial moments for different shapes Structural element properties are easily calculated or obtained from handbooks. Composite systems, such as reinforced concrete, should be based on the transformed section. Note that the structural

84 1-66 Special Features Structures/Fluid Flow/Thermal/Dynamics element formulation is a plane-stress formulation. If the element is representing a structure that is continuous in the direction perpendicular to the analysis plane (e.g., a concrete tunnel lining), the value specified for elastic modulus should be divided by (1 ν 2 ) to account for plane-strain conditions. Note that the mass density is required for the structural element formulation End Conditions and Applied Pressure The supplemental command STRUCT node n provides options for describing the end conditions of beam nodes. The options include (1) free or fixed x- andy-displacements or rotations; (2) applied velocities; and (3) applied loads or moments. These options are given by the following qualifying keywords following the node number n. fix fix fix <x> <y> <r> This option allows beam node n to have fixed x- and/or y-velocities or fixed angular velocities. free free free initial <x> <y> <r> This removes the constraint set by the fix keyword. condition is free.) keyword (The default Certain node variables can be assigned initial values. The following keywords apply. rvel value rotational velocity for beam nodes xdis value x-displacement for beam nodes

85 STRUCTURAL ELEMENTS 1-67 xvel value x-velocity for beam nodes ydis value y-displacement for beam nodes load yvel fx, fy, mom value y-velocity for beam nodes This allows the user to apply x- and/or y-direction forces or moments to node n for beams. The logic used in solving for forces on structural (beam) elements when applied as tunnel liners in saturated conditions is not capable of determining the fluid pressure difference between the inside and outside of the liner. The command STRUCT apply pressure p inside p outside allows the specification of the pressure difference so that the liner loads include hydrostatic forces. The variable p inside is the pressure inside the tunnel, and the variable p outside is the pressure at the block/liner interface. This pressure is applied to the block. If gravity is acting and the water density is specified, a pressure gradient may be applied to p inside and p outside. These values are taken as the pressure at a given reference location, and the hydrostatic gradient is superimposed. The command STRUCT apply ref loc xyis used to set the reference location. If the reference location is not given, no gradient is applied Summary of Commands Associated with Structural Elements All of the commands associated with structural elements are listed in Table 1.4. See Section 1 in the Command Reference for a detailed explanation of these commands.

86 1-68 Special Features Structures/Fluid Flow/Thermal/Dynamics Table 1.4 STRUCT Commands associated with structural elements keyword generate keyword (specify contacts) begin xy end xy max length value min length value mat n generate keyword (spray contacts) fang value mat n max length value min length value npoint np theta value xc value yc value generate keyword (optional) connect smat n stol value beam keyword begin xy end xy mat n beam keyword begin node n1 end node n1 mat n beam keyword table nt mat n node nid x y change keyword cons i jcons i jmat n mat n delete

87 STRUCTURAL ELEMENTS 1-69 Table 1.4 Commands associated with structural elements (continued) STRUCT keyword apply keyword pressure ref loc node nid keyword fix p inside p outside xy free initial load mat pin slave unslave keyword x y r keyword x y r keyword rvel xdis xvel ydis yvel fx fy m mat keyword x y m keyword x y value value value value value

88 1-70 Special Features Structures/Fluid Flow/Thermal/Dynamics Table 1.4 Commands associated with structural elements (continued) PROPERTY mat n keyword if cohesion if dilation if friction if kn if ks if tensile st area st density st inertia st pmtable st prat st rcrack st scresid st shape st spacing st thexp st thick st width st ycomp st yield st ymod st yresid value value value value value value value value value n value value value value value value value value value value value value

89 STRUCTURAL ELEMENTS 1-71 Table 1.4 Commands associated with structural elements (continued) PLOT stcon struct <keyword> afail avel axial ifail inormal interface ishear moment node number sdisp shear svel thick xdisp xvel ydisp yvel struct mat moment thrust thrust shear PRINT property struct struct keyword element interface node <keyword> <disp> <force> <geom> <keyword> <disp> <force> <state>

90 1-72 Special Features Structures/Fluid Flow/Thermal/Dynamics Example Application Inelastic Material Behavior of a Cantilever Beam A simple cantilever beam test is performed with a structural (beam) element, to illustrate the inelastic material behavior of the beam and compare the axial force and moment response of the beam to a failure envelope presented in a moment-thrust diagram. The beam is composed of a single beam segment that is fixed from translation and rotation at one end. A constant velocity in the axial direction and a constant rotational velocity are applied at the other end of the beam. The moment and axial force are monitored during the test. The beam material represents a 25 cm thick unreinforced shotcrete lining material. The cantilever beam specimen is 2.9 m long and 1 m wide. The properties of the material are density 2100 kg / m 3 Poisson s ratio 0.2 elastic modulus 30.5 GPa compressive yield strength MPa residual compressive strength 16.0 MPa tensile yield strength 2.2 MPa residual tensile strength 1.6 MPa Four simulations are run to illustrate the behavior of the inelastic material. In each case, a constant translational velocity and a rotational velocity are applied, and the inelastic material response is monitored and plotted on the P-M diagram. The translational velocity produces a compressive axial force in the beam. In the first case, the residual strengths are set equal to the initial strengths. In the second case, the actual residual strengths are specified. In the third case, the tensile yield strength is set to zero, and a crack depth ratio, h c, of 0.33 is prescribed. In the fourth case, a parabolic failure surface is input to define the inelastic material response. The data file for the four cases is listed in Example 1.5.

91 STRUCTURAL ELEMENTS 1-73 Example 1.5 Inelastic material behavior of a cantilever beam ;Project Record Tree export ;File:crliner.dat ;Title:Thrust-Moment Diagram ;Name:_liner ;Input:_syc/float/16.67e6/compressive strength ;Input:_scr/float/16.67e6/residual compressive strength ;Input:_syi/float/2.2e6/tensile strength ;Input:_syr/float/2.2e6/residual tensile strength ;Input:_rcrk/float/0.0/crack depth ratio ;Input:_pmtab/int/0/P-M diagram table def _liner ; geom ; thickness (height) _thi = 0.25 ; width _wid = 1.0 _moi = _wid*_thiˆ3/12. _are = _wid*_thi ; change sign of tensile strength (note: compression positive) _syim = -_syi _syrm = -_syr _scrp = _scr _Pc = _syc*_are _Mc = 0. _Pb = (_syim+_syc)*_are/2. _Mb = _moi*(_syc-_syim)/_thi _Pt = _syim*_are _Mt = 0. ; parabolic P-M diagram table _hpm = _Mb _kpm = _Pb _apm = -(_Pc-_Pb)ˆ2/_hpm _np = 100 _yinc = (_Pc-_Pt)/float(_np) loop n (0,_np) _y = _Pt + float(n)*_yinc _x = (_y-_kpm)*(_y-_kpm)/_apm + _hpm ytable(10,n+1) = _x ; M xtable(10,n+1) = -_y ; P endloop ; ; create diagram if _pmtab > 0

92 1-74 Special Features Structures/Fluid Flow/Thermal/Dynamics ; swap P and M for plotting purpose _ts = table_size(10) loop n (1,_ts) xtable(1,n) = ytable(10,n) ytable(1,n) = -xtable(10,n) endloop endif if _pmtab = 0 ns = 1 xtable(1,ns) = _Mc ytable(1,ns) = _Pc ns = ns + 1 xtable(1,ns) = _Mb ytable(1,ns) = _Pb if _rcrk > 0. ns = ns+1 _P4 = 0.5*_syc*_are*(1.-_rcrk) _M4 = _syc*_are*_thi*(1.+_rcrk-2.*_rcrkˆ2)/12. xtable(1,ns) = _M4 ytable(1,ns) = _P4 endif ns = ns + 1 xtable(1,ns) = _Mt ytable(1,ns) = _Pt endif end ; round edge block 0,0 0, , ,0 crack (0,0.05) (0.1,0.05) change mat 1 property mat 1 density 1E3 fix range 0,0.1 0,0.1 hide save crliner1.sav ; ; Case initial compressive and tensile strengths set _syc=16.67e6 _scr=16.67e6 _syi=2.2e6 _syr=2.2e6 _rcrk=0.0 set _pmtab=0 _liner struct beam begin 0.1,0.1 end 3.0,0.1 mat 1 prop mat 1 st_density 2.1E3 st_area _are st_inertia _moi st_thick _thi & st_ycomp _syc st_yield _syi st_yresid _syr st_scresid _scr st_rcra _rcrk & st_pmtable _pmtab st_ymod 30.5e9 st_prat 0.2 st_shape st_spac 1 & st_width 1 if_kn 1 if_ks 1

93 STRUCTURAL ELEMENTS 1-75 struct node 1 fix x fix y fix r struct node 2 fix x fix r struct node 2 init xvel -1e-7 init rvel 2e-6 save crliner2.sav call str.fin def _samp_pm _epnt = str_elem_head _P = -fmem(_epnt+$selfax) _M = fmem(_epnt+$selm2) _ns = _ns + 1 xtable(2,_ns) = abs(_m) ytable(2,_ns) = _P end def _load _nstep = 50 _np = 200 loop n (0,_np) command step _nstep endcommand _samp_pm endloop end set small=on _load save crliner3.sav ; restore crliner1.sav ; Case residual compressive and tensile strengths set _syc=16.67e6 _scr=16.0e6 _syi=2.2e6 _syr=1.6e6 _rcrk=0.0 set _pmtab=0 _liner struct beam begin 0.1,0.1 end 3.0,0.1 mat 1 prop mat 1 st_density 2.1E3 st_area _are st_inertia _moi st_thick _thi & st_ycomp _syc st_yield _syi st_yresid _syr st_scresid _scr st_rcra _rcrk & st_pmtable _pmtab st_ymod 30.5e9 st_prat 0.2 st_shape st_spac 1 & st_width 1 if_kn 1 if_ks 1 struct node 1 fix x fix y fix r struct node 2 fix x fix r struct node 2 init xvel -1e-7 init rvel 2e-6 save crliner4.sav call str.fin def _samp_pm _epnt = str_elem_head _P = -fmem(_epnt+$selfax) _M = fmem(_epnt+$selm2)

94 1-76 Special Features Structures/Fluid Flow/Thermal/Dynamics _ns = _ns + 1 xtable(2,_ns) = abs(_m) ytable(2,_ns) = _P end def _load _nstep = 50 _np = 200 loop n (0,_np) command step _nstep endcommand _samp_pm endloop end set small=on _load save crliner5.sav ; restore crliner1.sav ;Case crack depth ratio = 0.33 set _syc=16.67e6 _scr=16.67e6 _syi=0.0 _syr=0.0 _rcrk=0.33 set _pmtab=0 _liner struct beam begin 0.1,0.1 end 3.0,0.1 mat 1 prop mat 1 st_density 2.1E3 st_area _are st_inertia _moi st_thick _thi & st_ycomp _syc st_yield _syi st_yresid _syr st_scresid _scr st_rcra _rcrk & st_pmtable _pmtab st_ymod 30.5e9 st_prat 0.2 st_shape st_spac 1 & st_width 1 if_kn 1 if_ks 1 struct node 1 fix x fix y fix r struct node 2 fix x fix r struct node 2 init xvel -1e-7 init rvel 2e-6 save crliner6.sav call str.fin def _samp_pm _epnt = str_elem_head _P = -fmem(_epnt+$selfax) _M = fmem(_epnt+$selm2) _ns = _ns + 1 xtable(2,_ns) = abs(_m) ytable(2,_ns) = _P end def _load _nstep = 50 _np = 200 loop n (0,_np) command

95 STRUCTURAL ELEMENTS 1-77 step _nstep endcommand _samp_pm endloop end set small=on _load save crliner7.sav ; restore crliner1.sav ;Case input parabolic failure envelope set _syc=16.67e6 _scr=16.67e6 _syi=0.0 _syr=0.0 _rcrk=0.0 set _pmtab=10 _liner struct beam begin 0.1,0.1 end 3.0,0.1 mat 1 prop mat 1 st_density 2.1E3 st_area _are st_inertia _moi st_thick _thi & st_ycomp _syc st_yield _syi st_yresid _syr st_scresid _scr st_rcra _rcrk & st_pmtable _pmtab st_ymod 30.5e9 st_prat 0.2 st_shape st_spac 1 & st_width 1 if_kn 1 if_ks 1 struct node 1 fix x fix y fix r struct node 2 fix x fix r struct node 2 init xvel -1e-7 init rvel 2e-6 save crliner8.sav call str.fin def _samp_pm _epnt = str_elem_head _P = -fmem(_epnt+$selfax) _M = fmem(_epnt+$selm2) _ns = _ns + 1 xtable(2,_ns) = abs(_m) ytable(2,_ns) = _P end def _load _nstep = 50 _np = 200 loop n (0,_np) command step _nstep endcommand _samp_pm endloop end set small=on _load save crliner9.sav

96 1-78 Special Features Structures/Fluid Flow/Thermal/Dynamics In the first run, the initial compressive and tensile strength properties are assigned to the structural element material, and residual strengths are set equal to the initial strengths. Based upon equations Eqs. (1.29) through (1.31), the values for the three points of the moment-thrust diagram based upon the initial strengths are calculated to be at Point 1 P 1 = kN M 1 = 0 at Point 2 P 2 = 550.0kN M 2 = 0 at Point 3 P 3 = kN M 3 = 98.3kN Note that, by definition, compressive stresses are positive, so F c = and F t = Figure 1.33 plots the moment-thrust diagram for this case using these three points. The axial loads and moments calculated from the UDEC model are also shown. The crosses on the figure indicate the loading path that the UDEC model follows, and shows a good fit to the analytical results for the failure envelope. JOB TITLE :. UDEC (Version 5.00) (e+006) 4.50 LEGEND 11-Aug :06:53 cycle time 2.012E+04 sec table plot Analytical UDEC X X X Vs. 0.00E+00<X value> 9.83E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+005) Figure 1.33 Moment-thrust diagram at initial compressive and tensile strengths

97 STRUCTURAL ELEMENTS 1-79 For the second case, the actual residual compressive and tensile strengths are specified. In this simulation, when the stresses reach the initial ultimate strengths, the strength capacity is reduced to the residual value from that point on. Figure 1.34 displays the results for this case, and shows that the axial forces and moments are reduced to a cracked failure envelope. JOB TITLE :. UDEC (Version 5.00) (e+006) 4.50 LEGEND 11-Aug :35:41 cycle time 2.012E+04 sec table plot Analytical UDEC X X X Vs. 0.00E+00<X value> 9.83E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+005) Figure 1.34 Moment-thrust diagram include residual compressive and tensile strengths

98 1-80 Special Features Structures/Fluid Flow/Thermal/Dynamics For the third case, the effect of cracking is explicitly included by specifying a crack depth for the section. The initial and residual tensile yield strengths are set to zero (i.e., P 2 = 0), and a crack depth ratio, h c /h, of 0.33 is input. This results in a fourth point in the moment thrust diagram, for which the axial force, P 4, and moment, M 4, are calculated (using Eq. (1.33)) tobe at Point 4 P 4 = kN M 4 = 96.6kN Figure 1.35 compares the analytical results to the UDEC calculation and shows a good agreement for this case. JOB TITLE :. UDEC (Version 5.00) (e+006) 4.50 LEGEND 11-Aug :06:45 cycle time 2.012E+04 sec table plot Analytical UDEC X X X Vs. 0.00E+00<X value> 9.66E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+005) Figure 1.35 Moment-thrust diagram with zero tensile strength and crack depth ratio = 0.33

99 STRUCTURAL ELEMENTS 1-81 In the fourth case, a parabolic failure envelope is input to define the inelastic response. The equation of the parabola is defined to pass through the three points (Point 1, Point 2 and Point 3) as calculated for the first case. The parabolic equation has the form ( ) P 2 ] P3 M = M 3 [1 P 1 P 3 (1.34) Values for P and M that prescribe the failure envelope based on this equation are calculated in the FISH function liner, and stored in table 10. The failure envelope is plotted in the P-M diagram shown in Figure This figure also plots the UDEC results. This failure surface is invoked for the beam material in the UDEC model via the st pmtable property keyword. JOB TITLE :. UDEC (Version 5.00) (e+006) 4.50 LEGEND 11-Aug :40:54 cycle time 2.012E+04 sec table plot Analytical UDEC X X X Vs E+03<X value> 9.12E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+005) Figure 1.36 Moment-thrust diagram input P-M diagram table

100 1-82 Special Features Structures/Fluid Flow/Thermal/Dynamics Example Application Support of a Wedge in a Tunnel Roof A simple test is presented to demonstrate the capability of the structural element formulation to simulate failure and the post-failure strength of a tunnel lining. The example is a 6 m 6 m square tunnel containing a wedge in the roof. The wedge weighs 54 kn. The geometry is shown in Figure 1.37: JOB TITLE :. UDEC (Version 5.00) LEGEND 27-Jul :43:53 cycle time 2.278E+01 sec block plot structural elements plotted Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.37 Lined tunnel with wedge in roof The displacement of the wedge and the moments in the tunnel lining are compared for four conditions of the structural elements: (1) liner with high strength (elastic analysis); (2) liner with yield strength and residual strength equal to 18 MPa; (3) liner with yield strength equal to 18 MPa and residual strength equal to 16.7 MPa; and (4) double layer of lining with high strength (elastic analysis).

101 STRUCTURAL ELEMENTS 1-83 A rigid block analysis is performed; the material properties associated with the rock and joints are density 2700 kg / m 3 joint normal stiffness 10.0 GPa / m joint shear stiffness 10.0 GPa / m joint friction 30 A continuous liner is generated along the tunnel surface. The liner contains 240 structural nodes (specified by setting the keyword max = 0.1 in the STRUCT gen command). The properties of the liner are as follows. Liner Material Young s modulus 21.0 GPa Poisson s ratio 0.15 Case 1 Case 2 Case 3 tensile yield strength 40.0 MPa 18.0 MPa 18.0 MPa residual yield strength 40.0 MPa 18.0 MPa 16.7 MPa compressive yield strength 40.0 MPa 40.0 MPa 40.0 MPa Rock/Liner Interface normal stiffness shear stiffness friction 1.0 GPa / m 1.0 GPa / m 50 The UDEC data file is given in Example 1.6: Example 1.6 Lined tunnel with wedge in roof ; create tunnel with wedge block in roof round 0.05 edge 0.1 block -8,-8-8,8 8,8 8,-8 crack (-3,-8) (-3,8) crack (3,-8) (3,8) crack (-8,3) (8,3) crack (-8,-3) (8,-3) delete range -3,3-3,3 crack (-1,3) (1.5,8) crack (1,3) (-1.5,8) ; ; rock properties change mat 1

102 1-84 Special Features Structures/Fluid Flow/Thermal/Dynamics property mat 1 density 2.7E-3 group joint joint joint model area jks 1E4 jkn 1E4 jfriction 30 range group joint ; new contact default set jcondf joint model area jks=1e4 jkn=1e4 jfriction=30 fix range -8,8-8,0 fix range -8,8 5,8 save w1.sav ; ; structural liner and properties struct gen begin -3, end -3, max=0.1 mat 1 property mat 1 st_density 2.5E-5 st_prat 0.15 st_ycomp 40 st_yield 40 & st_ymod 2.1E4 st_yresid 40 st_area 0.1 st_inertia E-5 st_shape & st_spacing 1 st_thickness 0.1 st_width 1 if_friction 50 & if_cohesion 0 if_tensile 0 if_dilation 0 if_kn 1E3 if_ks 1E3 ; ; gravity load set gravity=0-10 ; history ydisplace 0.0,3.0 history yvelocity 0.0,3.0 ; ; Case 1 : elastic analysis solve step ratio 1.0E-7 save w2.sav ; ; Case 2 : yield strength = residual strength = 18 MPa property mat 1 st_density 2.5E-5 st_prat 0.15 st_ycomp 40 st_yield 18 & st_ymod 2.1E4 st_yresid 18 st_area 0.1 st_inertia E-5 st_shape & st_spacing 1 st_thick 0.1 st_width 1 if_friction 50 if_kn 1E3 & if_ks 1E3 solve ratio 1.0E-7 save w3.sav ; ; Case 3 : yield strength = 18 MPa residual strength = 16.7 MPa property mat 1 st_density 2.5E-5 st_prat 0.15 st_ycomp 40 st_yield 18 & st_ymod 2.1E4 st_yresid 16.7 st_area 0.1 st_inertia E-5 st_shape & st_spacing 1 st_thick 0.1 st_width 1 if_friction 50 if_kn 1E3 & if_ks 1E3 solve ratio 1.0E-7 save w4.sav ; Case 4 : elastic analysis with double layer of liner struct gen begin -3, end -3, max.1 mat 1 struct gen begin -3, end -3, max.1 mat 1 property mat 1 st_density 2.5E-5 st_prat 0.15 st_ycomp 40 st_yield 40 &

103 STRUCTURAL ELEMENTS 1-85 st_ymod 2.1E4 st_yresid 40 st_area 0.1 st_inertia E-5 st_shape & st_spacing 1 st_thickness 0.1 st_width 1 if_friction 50 & if_cohesion 0 if_tensile 0 if_dilation 0 if_kn 1E3 if_ks 1E3 ; ; gravity load set gravity=0-10 ; history ydisplace 0.0,3.0 history yvelocity 0.0,3.0 ; solve step ratio 1.0E-7 save w5.sav The results of the elastic analysis, Case 1, are compared to the analytic solution for bending of a beam with fixed ends. The analytical results are maximum displacement = m maximum bending moment = 36 kn-m The analytical solution does not include the weight of the liner, so the structural element density in the model is set to a very low value, to remove the influence of this weight on the UDEC solution. The UDEC results are plotted in Figure The maximum bending moment (35.0 kn-m) and maximum displacement (0.028 m) are very close to the analytical solution. Note that the sense of the structural moment plot is determined by the order in which the beam elements are generated. The sense of the moment plot can be changed by giving the yrev keyword switch. (See the PLOT command in Example 1.6.)

104 1-86 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE :. UDEC (Version 5.00) LEGEND Aug :25:15 cycle time 2.247E+01 sec block plot Moment on Structure Type # Max. Value struct E-02 displacement vectors maximum = 2.761E E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.38 Case 1: elastic solution The results for the Case 2 and Case 3 plastic analyses are shown in Figures 1.39 and 1.40, respectively. The structural nodes at the top corners of the lining fail in tension for both of these cases. (Use the PLOT struct afail command to verify the failure locations.) In Case 2, the maximum displacement is now greater, and the maximum moment smaller, than for the elastic case because of the plastic hinges that form at the corners. In Case 3, the displacement is even higher, and the moment lower, because of the residual strength at the plastic hinges. For Case 4, two layers of structural element lining are created, each with a thickness of 0.1 m. Figure 1.41 illustrates the two layers as if they have an actual thickness; the PLOT struct thick command produces this plot for visualization only. The structural nodes for each layer are actually at the same locations for the calculation. The result for this case is shown in Figure The maximum moment for each layer, and the maximum displacement, are approximately one third of the values calculated for the single layer (compare to Figure 1.38).

105 STRUCTURAL ELEMENTS 1-87 JOB TITLE :. UDEC (Version 5.00) LEGEND Aug :29:14 cycle time 4.169E+01 sec block plot Moment on Structure Type # Max. Value struct E-02 displacement vectors maximum = 3.631E E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.39 Case 2: yield strength = residual strength = 18 MPa JOB TITLE :. UDEC (Version 5.00) LEGEND 3-Aug :30:27 cycle time 4.103E+01 sec block plot Moment on Structure Type # Max. Value struct E-02 displacement vectors maximum = 4.202E E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.40 Case 3: yield strength = 18 MPa, residual strength = 16.7 MPa

106 1-88 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE :. UDEC (Version 5.00) LEGEND 27-Jul :50:42 cycle 0 time 0.000E+00 sec block plot structural elements plotted Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.41 Case 4: two layers of lining JOB TITLE :. UDEC (Version 5.00) LEGEND 3-Aug :15:49 cycle time 2.379E+01 sec block plot Moment on Structure Type # Max. Value struct E-02 struct E-02 displacement vectors maximum = 8.767E E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.42 Case 4: elastic solution for two layers of lining

107 STRUCTURAL ELEMENTS Example Application Circular Tunnel Excavation with Interior Support Conceptual understanding of the interaction between support loading and rock mass unloading is often explained in terms of a reaction curve for the rock medium and a stress-displacement curve for the support system. These curves are also called characteristic lines, characteristic curves or rock response curves. In the conceptual model used in this approach, the problem is reduced to consideration of a plane perpendicular to the tunnel axis, and all variables (i.e., stress, strain and displacement) vary only with radial distance from the tunnel. The ground reaction curve is frequently shown to consist of two parts: a descending portion and an ascending portion (as shown in Figure 1.43). The descending portion of the ground reaction curve generally consists of two distinct parts: an elastic part and an inelastic part (explained as follows). The model assumes that the excavation of the tunnel is simulated by quasi-statically unloading the boundary of the excavation. Upon unloading, the system responds elastically until the elastic limit is reached. Further unloading causes propagation of a plastic or failed zone around the excavation. If gravity is neglected, and the rock mass is assumed to behave as an isotropic, homogeneous, time-independent continuum, the descending portion of the ground reaction curve can be determined analytically based on material properties. For the example shown here, it is assumed that strength properties are sufficient, such that only elastic behavior need be considered. The expression for radial displacement in an elastic material is given by where r i,u i = tunnel radius and radial displacement; P o = in-situ isotropic stress (P o = P xx = P yy ); G = shear modulus; and = internal pressure. P i u i = r i 2G (P o P i ) (1.35) Expressions have also been developed to describe the support stiffness of a variety of supports, assuming that the support reaction is radially symmetric. For example, the stiffness of a blocked steel set is given by Daemen (1975) as 1 K ss = u i P i = Sr2 i EA + Sr4 i EI [ ] θ(θ+ sc) 2s Sr i θt B (1.36) A B E B

108 1-90 Special Features Structures/Fluid Flow/Thermal/Dynamics P yy Radial Deformation u i Excavated Profile Tunnel Profile P xx Support Pressure P i P xx P xx P yy P yy Support Pressure, P i Elastic ground response Elastic support response Figure 1.43 Radial Deformation, u i Conceptual representation of support reaction and ground reaction curves where K ss = stiffness of blocked steel set; A,E,I = steel cross-sectional area, elastic modulus and moment of inertia, respectively; S = steel set spacing; 2θ = angle between blocking points; n = π/θ = number of blocking points; s = sin θ; c = cos θ; E B, t B = elastic modulus and thickness of blocks; and A B = cross-sectional area of blocks.

109 STRUCTURAL ELEMENTS 1-91 Four calculations are made to establish a simple reaction curve for this example: (1) tunnel excavation without support (far-field stress constant); (2) tunnel excavation without support (far-field boundary fixed); (3) tunnel excavation with support (far-field stress constant); and (4) tunnel excavation with support (far-field boundary fixed). The two assumptions concerning far-field boundaries (i.e., constant stress and fixed) are required to bound the numerical solution because the analytical solution assumes infinite far-field boundaries. The problem parameters used to describe the ground reaction are r i = 1m G = 1MPa P o = 10 MPa The parameters used to describe the structural lining stiffness are A = 0.1 m 2 E = 2.57 GPa I = m 4 S = 1m n = 24 The blocking is described in the UDEC model by the interface normal stiffness, kn if (force / displacement), between the structure (i.e., steel set) and the rock. In this case, the last term in Eq. (1.36) is replaced by (2Sr i )/kn if, where kn if = A B E B t B (kn if = 1000 MN / m) The usual assumption made in analyzing blocked steel sets is that no shear force is transferred between the rock mass and the steel set. Consequently, friction and cohesion values were not specified (default value is zero). In setting up the numerical problem, the problem domain was divided into quadrant and concentric rings to facilitate zoning. All discontinuities were assigned as construction joints, using the JOIN CONTACT command. The net result is that the joints are essentially transparent, and do not affect the final result. The zone discretization is shown in Figure 1.44.

110 1-92 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE :. UDEC (Version 5.00) LEGEND Jul :29:58 cycle 0 time 0.000E+00 sec zones in fdef blocks block plot Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.44 Zoning for UDEC model of circular tunnel excavation The UDEC data file is given in Example 1.7: Example 1.7 Circular excavation with interior support round 0.01 edge 0.02 block -6,-6-6,6 6,6 6,-6 crack (-10,0) (10,0) crack (0,-10) (0,10) arc (0,0) (1,0) arc (0,0) (2,0) arc (0,0) (3.5,0) gen quad 0.8 range annulus (0,0) gen quad 0.6 range annulus (0,0) 2 1 gen edge 1.2 delete range annulus (0,0) 1 0 group zone block zone model elastic density bulk 2E3 shear 1E3 range group block join_cont boundary stress -10.0,0.0,-10.0 insitu stress -10.0,0.0,-10.0 history xdisplace 1.0,0.0 history xdisplace -1.0,0.0

111 STRUCTURAL ELEMENTS 1-93 history ydisplace 0.0,1.0 history ydisplace 0.0,-1.0 save support1.sav restore support1.sav title Circular Excavation with Interior Support - Unlined - Stress Boundary solve ratio 1.0E-5 save support2.sav restore support1.sav title Circular Excavation with Interior Support - Unlined - Fixed Boundary boundary xvelocity 0 boundary yvelocity 0 solve ratio 1.0E-5 save support3.sav restore support1.sav title Circular Excavation with Interior Support - Lined - Stress Boundary struct gen beg , end , max=1.0e9 min=0.02 & mat 1 property mat 1 st_density 2.5E-3 st_prat 0.2 st_ycomp 1E10 st_yield 1E10 & st_ymod 2.57E3 st_area 0.1 st_inertia E-5 st_shape & st_spacing 1 st_thickness 0.1 st_width 1 if_tensile 100 if_kn 1E3 if_ks 1 solve ratio 1.0E-5 save support4.sav restore support1.sav title Circular Excavation with Interior Support - Lined - Fixed Boundary struct gen beg , end , max=1.0e9 min=0.02 & mat 1 property mat 1 st_density 2.5E-3 st_prat 0.2 st_ycomp 1E10 st_yield 1E10 & st_ymod 2.57E3 st_area 0.1 st_inertia E-5 st_shape & st_spacing 1 st_thickness 0.1 st_width 1 if_tensile 100 if_kn 1E3 if_ks 1 boundary xvelocity 0 boundary yvelocity 0 solve ratio 1.0E-5 save support5.sav

112 1-94 Special Features Structures/Fluid Flow/Thermal/Dynamics The results of the analyses are compared in Figure As expected, the analytical solution falls between the numerical solutions obtained, assuming different boundary conditions. In analyzing the results, the internal pressure, P i, supplied by the support can be determined in one of two ways. The internal pressure is given by the thrust or axial force in the structural elements divided by the external radius of the support (i.e., 1 m), or by the radial force in the blocking divided by its tributary area (= 2π r i S/n). Both methods will yield the same value for P i supplied by the support. 10 Radial Support Pressure, Pi(Mpa) Numerical Solution (zero displacement boundary) Analytical Solution Numerical Solution (constant stress boundary) Support Reaction Figure Radial Displacement, Ui(mm) Comparison of ground reaction/support reaction lines

113 STRUCTURAL ELEMENTS Example Application Shotcrete Lined Tunnel Structural elements can be placed automatically around a tunnel periphery, even if the surface is highly irregular. The following example demonstrates this feature for the analysis of tunnel support in jointed rock. A tunnel is excavated in rock containing four joint sets. The orientations of the joint sets are summarized in Table 1.5: Table 1.5 Joint set orientations Joint Set Dip ( ) Trace Length (m) Gap Length (m) Spacing (m) mean/max.dev. mean/max.dev. mean/max.dev. mean/max.dev. Set / / / /0.1 Set / / / /0.1 Set / / / /0.1 Set / / / /0.1 The rock is assumed to be rigid relative to the joints. The material properties associated with the rock and joints are density 2500 kg/m 3 joint normal stiffness 1.0 GPa/m joint shear stiffness 1.0 GPa/m joint friction 45 For demonstration purposes, the tunnel is excavated instantaneously and lined with shotcrete. The shotcrete extends in a 300 arc, and covers all but the invert of the tunnel. Figure 1.46 shows the location of the shotcrete around the tunnel periphery.

114 1-96 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE : Shotcrete Lined Tunnel (*10^1) UDEC (Version 5.00) LEGEND Oct :30:02 cycle 0 time 0.000E+00 sec block plot structural elements plotted Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 1.46 Shotcrete applied in 300 arc on tunnel periphery The properties of the shotcrete are as follows. Shotcrete density 2500 kg/m 3 Young s modulus 21.0 GPa Poisson s ratio 0.15 tensile yield strength 20.0 MPa residual yield strength 10.0 MPa compressive yield strength 40.0 MPa Rock/Shotcrete Interface normal stiffness shear stiffness friction 1.0 GPa/m 1.0 GPa/m 45 The data file for this example is given in Example 1.8.

115 STRUCTURAL ELEMENTS 1-97 Example 1.8 Shotcrete lined tunnel round 0.01 edge 0.02 set random 1000 block 0,0 0,10 10,10 10,0 jset angle 45,5 trace 20,2 gap 1,0.1 spacing 0.8,0.1 origin 0,0 jset angle 25,5 trace 10,1 gap 1,0.1 spacing 0.6,0.1 origin 0,0 jset angle 135,5 trace 5,0.2 gap 0.2 spacing 0.7,0.1 origin 0,0 jset angle 95,5 trace 2.5,0.2 gap 0.2 spacing 0.5,0.1 origin 0,0 crack (3.5,2.7) (6.8,2.7) jdelete delete range 3,7 2.7,5 delete range annulus (5,5) 0 2 delete area change mat 1 property mat 1 density 2.5E3 group joint joint joint model area jks 1E9 jkn 1E9 jfriction 45 range group joint ; new contact default set jcondf joint model area jks=1e9 jkn=1e9 jfriction=45 fix range 0,0.5 0,10 fix range 9.5,10 0,10 fix range 0,10 0,0.5 fix range 0,10 9.5,10 insitu stress ,0.0, xgrad 0.0,0.0,0.0 ygrad & 0.0,0.0, set gravity=0-10 ; shotcrete lining struct gen beg , end , max=1.0e9 min=0.02 mat 1 property mat 1 st_dens 2.5E3 st_prat 0.15 st_ycomp 100e6 st_yield 100e6 & st_ymod 2.1E10 st_yresid 100e6 st_area 0.1 st_inert E-5 st_shape & st_spacing 1 st_thickness 0.1 st_width 1 if_frict 45 if_kn 1E9 & if_ks 1E9 save stun1.sav solve ratio 1.0E-5 save stun2.sav ; ; change shotcrete properties to actual values with residual str = 20 MPa property mat 1 st_density 2.5E3 st_prat 0.15 st_ycomp 40e6 st_yield 20e6 & st_ymod 2.1E10 st_yresid 20e6 st_area 0.1 st_inertia E-5 st_shape & st_spacing 1 st_thick 0.1 st_width 1 if_friction 45 if_kn 1E9 & if_ks 1E9 solve ratio 1.0E-6 save stun3.sav

116 1-98 Special Features Structures/Fluid Flow/Thermal/Dynamics ; rest stun2.sav ; change shotcrete properties to actual values with residual str = 10 MPa property mat 1 st_density 2.5E3 st_prat 0.15 st_ycomp 40e6 st_yield 20e6 & st_ymod 2.1E10 st_yresid 10e6 st_area 0.1 st_inertia E-5 st_shape & st_spacing 1 st_thick 0.1 st_width 1 if_friction 45 if_kn 1E9 & if_ks 1E9 solve ratio 1.0E-6 save stun4.sav Note that, for this example, it is not necessary to specify the shape factor and moment of inertia for the shotcrete because it has a uniform cross section. These factors are calculated automatically. When the STRUCT gen begin... end... command is issued, nodes are created automatically in order to follow the irregular surface of the excavation. Two cases are evaluated. First, a simulation is made with the residual yield strength of the shotcrete set equal to the initial yield strength of 20 MPa. Figure 1.47 shows the axial force distribution in the shotcrete, and Figure 1.48 shows the locations along the lining that have failed in tension for this case. The moment-thrust diagram for the shotcrete is shown in Figure Note that in this figure, three failure surfaces are plotted, for safety factors of 1.0, 1.2 and 1.4. The axial load and moment calculated for each liner segment are also denoted in this figure by asterisk markers. A few asterisks are plotted at or near the failure surface corresponding to a factor of safety equal to 1, indicating segments at or near failure. In the second simulation, the residual strength is reduced to 10 MPa. Figure 1.50 shows that axial forces in the liner are reduced, as compared to Figure 1.47.

117 STRUCTURAL ELEMENTS 1-99 JOB TITLE : Shotcrete Lined Tunnel (*10^1) UDEC (Version 5.00) LEGEND Oct :25:57 cycle 7390 time 1.793E+00 sec block plot Axial Force on Structure Type # Max. Value struct E+05 structural elements plotted Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure (*10^1) Axial force distribution in shotcrete lining residual yield strength = 20 MPa JOB TITLE : Shotcrete Lined Tunnel (*10^1) UDEC (Version 5.00) LEGEND Oct :25:57 cycle 7390 time 1.793E+00 sec block plot Structural Element Failure Axial yielding structural elements plotted Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure (*10^1) Tensile failure locations in shotcrete residual yield strength = 20 MPa

118 1-100 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE : Shotcrete Lined Tunnel UDEC (Version 5.00) (e+006) 5.00 LEGEND 14-Oct :25:57 cycle 7390 time 1.793E+00 sec moment thrust diagram Safety Factors 1.0,1.2, Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+004) Figure 1.49 Moment-thrust diagram for tensile yield strength = 20 MPa and compressive yield strength = 40 MPa JOB TITLE : Shotcrete Lined Tunnel (*10^1) UDEC (Version 5.00) LEGEND Oct :25:44 cycle 7380 time 1.790E+00 sec block plot Axial Force on Structure Type # Max. Value struct E+05 structural elements plotted Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure (*10^1) Axial force distribution in shotcrete lining residual yield strength = 10 MPa

119 STRUCTURAL ELEMENTS Example Application Slope Stabilization In this example, a jointed rock slope is stabilized by applying a shotcrete lining to the slope face. The problem geometry, shown in Figure 1.51, consists of a slope cut in a rock containing two continuous joint sets. JOB TITLE :. UDEC (Version 5.00) (*10^1) LEGEND 12-Aug :55:29 cycle 660 time 1.255E-01 sec block plot Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 1.51 Slope cut in a jointed rock The properties of the rock and joints are density 2500 kg/m 3 rock bulk modulus GPa rock shear modulus 10.0 GPa joint normal stiffness 10.0 GPa/m joint shear stiffness 10.0 GPa/m joint friction 15 The slope is not stable for these conditions. This can be seen from the displacement vector plot in Figure This result was obtained by first solving for the equilibrium state with a high joint friction angle, and then reducing the friction to the actual value.

120 1-102 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE :. UDEC (Version 5.00) (*10^1) LEGEND 12-Aug :55:35 cycle time 2.026E+00 sec block plot displacement vectors maximum = 1.112E E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 1.52 Unsupported slope is unstable A slope lining can be placed on the slope using the structural element logic in UDEC. The lining is 0.1m thick shotcrete with the following properties. Shotcrete density 2500 kg/m 3 Young s modulus 20.0 GPa Poisson s ratio 0.15 tensile yield strength 3.0 MPa residual yield strength 0.0 MPa compressive yield strength 30.0 MPa Rock/Shotcrete Interface normal stiffness GPa/m shear stiffness GPa/m cohesive bond strength 2.0 MPa tensile bond strength 1.0 MPa friction 0 The lining is added at the equilibrium state with high joint friction. Now, when the friction is reduced, the slope is stable. This can be seen from Figure 1.53:

121 STRUCTURAL ELEMENTS JOB TITLE :. UDEC (Version 5.00) (*10^1) LEGEND 12-Aug :55:37 cycle 4481 time 5.756E-01 sec block plot displacement vectors maximum = 2.691E E -3 Axial Force on Structure Type # Max. Value struct E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 1.53 Slope is stabilized with shotcrete lining The data file for this example is given in Example 1.9: Example 1.9 Slope stabilization round 0.05 edge 0.1 block 0,-5 0,0 5,0 11,10 22,10 22,-5 jset angle 20 spacing 2 origin 5,1 jset angle 80 spacing 3 origin 5,0 delete area 0.1 delete range 10.5,12 9.5,10 gen edge 10.0 group zone block zone model elastic density 2.5E-3 bulk E4 shear 1E4 range group & block group joint joint joint model area jks 1E4 jkn 1E4 jfriction 45 range group joint ; new contact default set jcondf joint model area jks=1e4 jkn=1e4 jfriction=45 insitu stress ,0.0,-0.25 xgrad 0.0,0.0,0.0 ygrad ,0.0,0.025 boundary xvelocity 0 range -0.1, ,0.1 boundary xvelocity 0 range 21.9, ,10.1 boundary yvelocity 0 range -0.1, ,-4.9

122 1-104 Special Features Structures/Fluid Flow/Thermal/Dynamics set gravity=0-10 history xdisplace 6.0,2.0 history xdisplace 9.0,7.0 history ydisplace 6.0,2.0 history ydisplace 9.0,7.0 ; slope stable solve ratio 1.0E-5 save ls1.sav ; slope fails at 15 degree friction group joint weak joint group joint weak joint range group joint ; changed name joint model area jks 1E4 jkn 1E4 jfriction 15 range group weak joint ; new contact default set jcondf joint model area jks=1e4 jkn=1e4 jfriction=15 cycle save ls2.sav ; slope staabilized with liner group joint weak joint group joint weak joint range group joint ; changed name joint model area jks 1E4 jkn 1E4 jfriction 15 range group weak joint ; new contact default set jcondf joint model area jks=1e4 jkn=1e4 jfriction=15 ; struct gen begin , E-5 end , max=0.5 min=0.1 & mat 1 property mat 1 st_density 2.5E-3 st_prat 0.15 st_ycomp 30 st_yield 3 & st_ymod 2E4 st_yresid 3 st_area 0.1 st_inert E-5 st_shape & st_spac 1 st_thickness 0.1 st_width 1 if_cohesion 2 if_tensile 1 if_kn & 1E5 if_ks 1E5 solve ratio 1.0E-5 save ls3.sav

123 STRUCTURAL ELEMENTS Support Members (SUPPORT Command) Support members are intended to model hydraulic props, wooden props, sticks or wooden packs. In its simplest form, a support member is a spring connected between two boundaries. The spring may be linear, or it may obey an arbitrary relation between axial force and axial displacement, as prescribed from a table of values. The support member has no independent degrees of freedom; it simply imposes forces on the boundaries to which it is connected. A support member may also have a width associated with it. In this case, it behaves as if it were composed of several parallel members spread out over the specified width. The force-displacement behavior can also be made load-rate dependent, as described in Section * Standard Formulation In the standard formulation for support elements, two options are available to describe the axial force-displacement relation. In one option, the relation is elastic-plastic and is defined by an axial stiffness and a compressive yield limit. In the other option, the relation between force and displacement is prescribed by a table of force/displacement values. These two options are illustrated in Figure 1.54: axial force axial force F y k a unload axial displacement axial displacement (a) elastic-plastic (b) specified by TABLE Figure 1.54 Force-displacement behavior for standard support model * The load-rate dependent model was developed with funding and technical support from CSIR MiningTEK, Johannesburg, Republic of South Africa. The model is experimental and has only been partially tested. It should be used with caution.

124 1-106 Special Features Structures/Fluid Flow/Thermal/Dynamics The formulations for the two options differ in their relation of axial force to axial displacement. For the elastic-plastic option, the force/displacement relation is incremental; for the table option, the force in the support member is related to the total displacement of the member. When a support member has nonzero width (i.e., it is divided into sub-members), the force in each sub-member is computed in one of two ways. For the elastic-plastic option, the force F in each sub-member is defined by and F = F k a (n + 1) u a (1.37) F y (n + 1) where n = number of sub-members; u = axial displacement increment of the sub-member; F = axial force increment; k a = axial stiffness; and = compressive force limit. F y and for the table relation, the force is defined by F = f (u) (n + 1) (1.38) where u = total displacement of the sub-members; and f (u) = table look-up function. The total force exerted by the support is the sum of the sub-member forces. These formulations also differ in their treatment of unloading. The elastic-plastic option unloads using the axial stiffness, k a. The relation specified by a table unloads according to the specified table, as indicated in Figure 1.54.

125 STRUCTURAL ELEMENTS Load-Rate Dependency Several types of support elements used in reef mining (e.g., profile props, yielding props and sticks) have a force-displacement behavior that is dependent on the rate of loading. Measurements in the laboratory and underground indicate that the shape of the force-displacement curve is similar for different loading rates, but that the maximum force that the support can withstand varies with the loading rate. The force-displacement behavior, including load-rate dependency, is considered to be composed of a rate-dependent curve superimposed on a static (slow loading rate) curve. The additional force due to the higher loading rate is calculated incrementally, based upon an increment in deformation and the proximity of the support force to the maximum force level. This is expressed as [ F i+1 = F i + C 1 F ] i (u i+1 u i ) (1.39) F max where F i+1 = new addition in support force; F i = old addition in support force; u i+1 = new deformation of support element; u i = old deformation of support element; C = stiffness constant; and F max = maximum support force addition. The total force in the support element is calculated by adding the rate-dependent force increment to the static support force: F total = F static + F i+1 (1.40) As the total rate-dependent force increment, F, approaches the maximum force increment, F max, the increment in total support force is gradually reduced, and an asymptotically increasing support force is added to the static force. This ensures that the shape of the static force-displacement curve is replicated for higher loading rates. The stiffness constant, C, controls the rate at which the maximum force level is approached. The maximum force increment, F max, is estimated for a specific loading rate. The following expression is used to describe the maximum force. F max = u α F max F max.static (1.41) where F max.static u F max α = maximum static force; = loading rate; = maximum force at highest loading for the support element; and = constant depending on support type.

126 1-108 Special Features Structures/Fluid Flow/Thermal/Dynamics For large displacements of a prop, the support force will eventually go to zero as the prop fails. This effect is simulated in UDEC, by having F max decrease linearly for displacements larger than the displacement at which the maximum static force, F max.static, is reached. The maximum support force addition for displacements greater than this limiting displacement is F max.lim = F max K F (u u lim ) (1.42) where F max.lim = maximum support force addition for u>u lim ; u = displacement of the support unit; u lim = displacement at the maximum static force; and = constant for reducing maximum support force addition. K F K F defines the gradient from the point at which u lim is reached to the point that F max is equal to zero. Figure 1.55 illustrates the force-displacement behavior that is represented by Eqs. (1.39) through (1.42). support force rate-dependent force-displacement curve F max F total C ΔF F static F max. static ΔF max static force-displacement curve (specified via TABLE) Figure 1.55 u u lim support displacement Force-displacement behavior including load-rate dependent support force

127 STRUCTURAL ELEMENTS The support stiffness, C, ineq. (1.39) can also be related to the maximum support force addition. The following relation is used in UDEC. where k = constant depending on support type. C = k( F max + F max.static ) (1.43) A tensile limit can also be specified for the model. This is the maximum tensile force that the support element can withstand. If the tensile force in the support exceeds t max, the support force is set to zero and the support is deleted. The constant k, along with the parameters F max, α and t max, and a table defining the static force-displacement relation, are the required input for the rate-dependent model Numerical Stability For support members, the stiffness is not taken into account in the consideration of numerical stability, mainly because it is difficult to estimate the stiffness in advance for table look-ups. If the support is stiffer than the rock, this may lead to numerical instability, but the reverse is likely to be the case in most problems. If numerical instabilities do occur, the timestep should be reduced using the FRACTION command Support Member Properties The standard model for support elements in UDEC requires two input parameters: (1) axial stiffness of the support member, k a (force/displacement); and (2) compressive yield strength (force) of the support member, F y. If the support member contains sub-elements, then the axial stiffness and yield strength are for the group of sub-members. Alternatively, the relation between axial force and axial displacement can be specified by a look-up table. However, a table is not recommended if the standard support is subjected to significant unloading. When a table is to be specified, the axial stiffness, k a, should be entered as a negative integer. The table number is the absolute value of this integer. Table values are specified with the TABLE command. For the load-rate dependent model, the user must supply four model parameters plus a table of the static force-displacement relation. The four parameters are (1) exponent constant, α; (2) maximum support force addition, F max ; (3) support stiffness constant, k; and

128 1-110 Special Features Structures/Fluid Flow/Thermal/Dynamics (4) tensile strength (force), t max. The four parameters are usually determined from laboratory or underground testing. CSIR (1993) provides some preliminary results for k, F max and α for three support types. The values are summarized in Table 1.6: Table 1.6 Support properties for profile props, sticks and yielding props (CSIR 1993) Support Type α F max (kn) k loading rate (m/s) profile props sticks yielding props The force-displacement behavior specified by a table for the rate-dependent model must be concave, and must end at a zero support force. Support elements typically offer almost no resistance to tensile loading. However, it is advisable to specify a small tensile strength, t max, for a support element instead of setting the strength to zero. This ensures that the element is not deleted accidentally during the initial transient phase of the calculation Summary of Commands Associated with Support Members All of the commands associated with support members are listed in Table 1.7. See Section 1 in the Command Reference for a detailed explanation of these commands.

129 STRUCTURAL ELEMENTS Table 1.7 Commands associated with support members SUPPORT xy keyword angle delete mat remove segment type width PROPERTY mat np keyword sup alpha sup constant sup fmax sup kn sup kn sup tmax sup ycomp value np ns keyword standard rate dependent value value value value value -nt value value TABLE nt x1 y1 <x2 y2> <x3 y3>... PLOT support PRINT property support support SET sup delete

130 1-112 Special Features Structures/Fluid Flow/Thermal/Dynamics Example Application Support of Faulted Ground This example problem illustrates the use of support members in faulted ground. Figure 1.56 shows the location of the vertical fault and support members. The support members have the forcedisplacement relation shown in Figure The specified support yields at 40 MN, as shown. JOB TITLE :. UDEC (Version 5.00) LEGEND 28-Jul :36:21 cycle 0 time 0.000E+00 sec Support Element Locations block plot Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.56 Support members before loading 40 Force (x10 6 N) Displacement (m) Figure 1.57 Force-displacement relation for support in example problem

131 STRUCTURAL ELEMENTS Example 1.10 contains the data file for this example. Example 1.10 Support of faulted ground round 0.1 edge 0.2 block -1,0-1,7 9,7 9,0 crack (0,0) (0,7) crack (8,0) (8,7) crack (0,1) (8,1) crack (0,3) (8,3) crack (4,3) (4,7) delete range 0,8 1,3 gen quad 1.1 range group zone block zone model mohr density 1E3 bulk 1.5E8 shear 5E7 coh 2.5E5 range group & block change mat 1 property mat 1 density 1E3 group joint joint joint model area jks 1E8 jkn 1E8 range group joint ; new contact default set jcondf joint model area jks=1e8 jkn=1e8 fix range -1,0 0,7 fix range 8,9 0,7 boundary xvelocity 0 range -0.1, ,0.1 boundary yvelocity 0 range -0.1, ,0.1 hide 4,8 3,7 boundary xvelocity 0 range -0.1, ,7.1 boundary yvelocity 0 range -0.1, ,7.1 show hide 0,4 3,7 boundary yvelocity -0.2 range 3.9, ,7.1 show save sup1.sav ; support 4 2 wid 3 seg 20 mat 3 prop mat 3 sup_kn -1 table e e7 cycle 5000 save sup2.sav Figure 1.58 shows the deformed position of the supports after the upper surface on the right side of the model has displaced downward approximately 0.6 m.

132 1-114 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE :. UDEC (Version 5.00) LEGEND 28-Jul :33:20 cycle 5000 time 2.654E+00 sec Support Element Locations block plot Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.58 Support members after loading

133 STRUCTURAL ELEMENTS Example Application Load-Rate Dependent Support The rate-dependent support model is used to simulate the behavior of profile props subjected to loading and unloading. Each prop is composed of five sub-elements and has a width of 2 m. The model configuration is shown in Figure 1.59: JOB TITLE :. UDEC (Version 5.00) LEGEND 28-Jul :34:02 cycle 0 time 0.000E+00 sec block plot Support Element Locations Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.59 Model test for rate-dependent support members There are two stages to this analysis. First, the model is subjected to gravity loading and brought to an equilibrium state. Then, the bottom-right triangular block is freed. The prop connected to this block unloads when this block moves. The axial force and displacement in the props are monitored during both stages. Example 1.11 contains the data file for this example. Example 1.11 Load-rate dependent support round 0.1 edge 0.2 block -1,0-1,7 9,7 9,0 crack (0,0) (0,7) crack (8,0) (8,7) crack (0,1) (8,1) crack (0,3) (8,3) crack (8,1) (9,1) crack (4.5,1) (8,0)

134 1-116 Special Features Structures/Fluid Flow/Thermal/Dynamics delete range 0,8 1,3 delete range 8,9 0,1 change mat 1 property mat 1 density 1E3 ; ; assign friction for vertical and inclined joints group joint joint10 group joint joint5 range -0.1, ,7.1 group joint joint5 range 7.9, ,7.1 joint model area jks 1E8 jkn 1E8 jfriction 10 range group joint10 joint model area jks 1E8 jkn 1E8 jfriction 5 range group joint5 ; new contact default set jcondf joint model area jks=1e8 jkn=1e8 jfriction=10 fix range -1,0 0,7 fix range 8,9 0,7 fix range 0,8 0,1 insitu stress ,0.0, set gravity=0-10 save lrsup1.sav ; install two supports sup 3 2 seg=5 wid=2.0 mat=3 type rate_dep sup 6 2 seg=5 wid=2.0 mat=3 type rate_dep prop m=3 sup_alfa= sup_fmax= sup_cons=27.6 sup_tmax=1000 prop m=3 sup_kn=-1 table e e e-3 0 ; call support.fin def sup_forc1 ; force/disp. hist of non-deleted support : address = 2344 ; force/disp. hist of deleted support : address = 2489 sup_forc1 = fmem(iad_sup_1+$ksufn) sup_disp1 = fmem(iad_sup_1+$ksuun) sup_forc2 = fmem(iad_sup_2+$ksufn) sup_disp2 = fmem(iad_sup_2+$ksuun) end set iad_sup_1=2344 iad_sup_2=2489 sup_forc1 history ncyc 1 history sup_forc1 history sup_disp1 history sup_forc2 history sup_disp2 ; ; equilibrate solve ratio 1.0E-5

135 STRUCTURAL ELEMENTS save lrsup2.sav ; ; allow support to be deleted set sup_del free range 4.5,8 0,1 cycle 5000 save lrsup3.sav The load/displacement response of one sub-member of the left prop during the gravity-loading stage is shown in Figure The response of the right prop is similar. The force-displacement behavior is dependent on the prescribed properties for a profile prop, taken from Table 1.6. The initial loading of the prop is nonlinear, followed by linear unloading and reloading as the force in the sub-member approaches a static value of approximately 32 kn. JOB TITLE :. UDEC (Version 5.00) LEGEND 28-Jul :18:44 cycle 151 time 2.288E-01 sec history plot Y-axis: 1 - Fish: sup_forc1 X-axis: 2 - Fish: sup_disp1 (e+004) Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-003) Figure 1.60 Axial force versus displacement response for left prop When the floor block is freed in the second stage, it begins to slide and the right prop unloads. By specifying the command SET sup del, the support will be deleted when the tensile force exceeds the specified tensile limit, sup tmax (PROPERTY command). The unloading of the right prop is indicated by the load/displacement plot in Figure The load is transferred to the left prop, as shown in Figure 1.62.

136 1-118 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE :. UDEC (Version 5.00) (e+004) 4.00 LEGEND 28-Jul :21:23 cycle 5151 time 7.806E+00 sec history plot Y-axis: 3 - Fish: sup_forc2 X-axis: 4 - Fish: sup_disp Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-003) Figure 1.61 Axial force versus displacement response for right prop unloading JOB TITLE :. UDEC (Version 5.00) (e+004) 8.00 LEGEND 28-Jul :21:23 cycle 5151 time 7.806E+00 sec history plot Y-axis: 1 - Fish: sup_forc1 X-axis: 2 - Fish: sup_disp Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-002) Figure 1.62 Axial force versus displacement response in left prop for right prop unloading

137 STRUCTURAL ELEMENTS Material Properties Property numbers are assigned to local reinforcement, cables, structural (beam) elements and support members with the PROPERTY command, and to rockbolts with the STRUCT property command. Note that all quantities must be given in an equivalent set of units (see Table 1.8). Note that for reinforcement elements, cable elements, rockbolts and support members, stiffness has units of [force/displacement] and strength has units of [force]. For structural (beam) elements, stiffness has units of [stress/displacement] and strength has units of [stress]. Also, for cable elements, rockbolts and structural (beam) elements, the code does take into account the weight of the structure when calculating loads. Table 1.8 Systems of units structural elements Property Unit SI Imperial Area length 2 m 2 m 2 m 2 cm 2 ft 2 in 2 Bond Stiffness force/length/disp N/m/m kn/m/m MN/m/m Mdynes/cm/cm lbf/ft/ft lbf/in/in Bond Strength force/length N/m kn/m MN/m Mdynes/cm lbf/ft lbf/in Density mass/volume kg/m kg/m kg/m g/cm 3 slugs/ft 3 snails/in 3 Elastic Modulus stress Pa kpa MPa bar lbf/ft 2 psi Failure Strain Moment of Inertia length 4 m 4 m 4 m 4 cm 4 ft 4 in 4 Plastic Moment force-length N-m kn-m MN-m Mdynes-cm ft-lbf in-lbf Stiffness* force/disp N/m kn/m MN/m Mdynes/cm lbf/ft lbf/in Stiffness** stress/disp Pa/m kpa/m MPa/m bar/cm lbf/ft 3 lbf/in 3 Yield Strength* force N kn MN Mdynes lbf lbf Yield Strength** stress Pa kpa MPa bar lbf/ft 2 psi where 1 bar = 10 6 dynes / cm 2 =10 5 N/m 2 =10 5 Pa, 1 atm = bars = 14.7 psi = 2116 lb f /ft 2 = Pa, 1 slug = 1 lb f s 2 / ft = kg, 1 snail = 1 lb f s 2 / in, and 1 gravity = 9.81 m / s 2 = 981 cm / s 2 = ft / s 2. * Refers to axial, normal and shear stiffnesses and strength related to the material of reinforcement elements, cable elements, rockbolts or support members. ** Refers to axial, normal and shear stiffnesses and strength related to the material of structural (beam) elements and structure/block interfaces.

138 1-120 Special Features Structures/Fluid Flow/Thermal/Dynamics 1.5 Modeling Considerations D/3D Equivalence Reducing 3D problems to 2D problems with regularly spaced structural elements involves averaging the effect in 3D over the distance between the elements. Donovan et al. (1984) suggest that linear scaling of material properties is a simple and convenient way of distributing the discrete effect of elements over the distance between elements in a regularly spaced pattern. The relation between actual properties and scaled properties can be demonstrated by considering the strength properties for regularly spaced rockbolts. The actual maximum normal force per length of the rockbolt is defined by Eq. (1.25). Internally, UDEC uses the expression (Fn max ) s L = (cs ncoh ) s + p tan(cs nfric ) (perimeter) s (1.44) where (Fn max ) s is the (scaled) maximum normal force per unit model thickness calculated by UDEC. (The superscript s does not denote a power.) We want the total force calculated by UDEC over a spacing, S, to be the same as the actual force. The actual maximum normal force is then F max n = (F max n ) s S (1.45) and the actual normal force is F n = (F n ) s S (1.46) The relation between the actual force and the UDEC force can be satisfied by substituting Eq. (1.45) and the following relations into Eq. (1.44). (cs ncoh ) s = cs ncoh S (perimeter) s = perimeter S (1.47) (1.48)

139 STRUCTURAL ELEMENTS The actual normal stress on the rockbolt, σ n, is calculated by dividing the actual force by the actual effective area (perimeter L): σ n = (F n) s S perimeter L (1.49) Note that the choice to scale perimeter is arbitrary, because only the product tan(cs nfric ) perimeter is relevant. Alternatively, the friction term could be scaled. It is important to remember that the forces (and moments) for structural elements that are calculated by UDEC are scaled forces (and moments). The actual forces and moments can be calculated by multiplying the UDEC forces and moments by S. FISH access to UDEC values for forces and moments access scaled values, and thus should be multiplied by the appropriate spacing value to determine the actual values. A scaling property is provided for the structural elements to scale properties, and account for a spaced pattern of structural elements: reinforcement (r spacing), cables (cb spacing), rockbolts (spacing), beams (st spacing) and supports (sup spacing). When the spacing property is specified, the actual properties of the structural element are input. The scaled properties are then calculated automatically by dividing the actual properties by the spacing, S. When the calculation is complete, the actual forces and moments in the spaced structural elements are then determined automatically (by multiplying by the spacing) for presentation in output results. The following lists summarize the structural element properties that are scaled when the spacing property is specified to simulate regularly spaced structural elements. For reinforcement elements, several properties are scaled: (1) axial stiffness; (2) ultimate axial capacity; (3) shear stiffness; and (4) ultimate shear capacity. For cable elements, several properties are scaled: (1) elastic modulus of the cable; (2) tensile yield strength of the cable; (3) compressive yield strength of the cable; (4) stiffness of the grout; and (5) cohesive strength of the grout.

140 1-122 Special Features Structures/Fluid Flow/Thermal/Dynamics For rockbolt elements, several properties are scaled: (1) elastic modulus of the rockbolt; (2) plastic moment of the rockbolt; (3) tensile yield strength of the rockbolt; (4) compressive yield strength of the rockbolt; (5) stiffness of the shear coupling spring; (6) cohesive strength of the shear coupling spring; (7) stiffness of the normal coupling spring; (8) cohesive strength of the normal coupling spring; and (9) exposed perimeter of the rockbolt. For beam elements, several properties are scaled: (1) elastic modulus of the element; (2) tensile yield strength of the element; (3) residual tensile yield strength of the element; (4) cohesive yield strength of the element; (5) residual compressive yield strength of the element; (6) interface normal stiffness; (7) interface shear stiffness; (8) interface cohesion; and (9) interface tensile strength. For support elements, two properties are scaled: (1) axial stiffness of the support member; and (2) compressive yield strength of the support member. The spacing property also applies to gravity loads, which are calculated using the cross-sectional area and the scaled structure density. Any pretensioning that is specified to cable elements (using the optional value preten with the CABLE command) is scaled when cb spacing is given. If loading is applied using the STRUCT node n load command, these loads are not scaled when the spacing property is assigned. The loads should be scaled by dividing by S.

141 STRUCTURAL ELEMENTS The following example illustrates the simulation of regularly spaced structural elements. In this case, vertical rockbolts at an equal spacing of 2 m are subjected to axial loading. The actual elastic modulus of the rockbolt is 10 GPa, and the actual stiffness of the shear coupling spring is 1 GN/m/m. The cohesive strength of the shear coupling spring is set to a high value to prevent shear failure for this simple example. A vertical axial loading of 2 MN is applied at the top of the pile, and the pile spacing is set to 2 m. Example 1.12 lists the commands for this example. The model is run for both the case in which spacing is specified, and the case in which it is not. In the second case, the input values for elastic modulus and shear coupling spring stiffness are scaled (by dividing by 2). Note that for both cases, the applied vertical load is scaled (STRUCT node 2 load 0.0, , 0.0). Figure 1.63 displays the result for the first case. When spacing is given, the actual axial forces are displayed in the pile axial force plot. Figure 1.47 shows the results for the second case. When spacing is not given, but the input properties are scaled, the axial force plot displays the scaled values for axial force. The axial forces in Figure 1.64 must be multiplied by 2 to obtain the actual values. Example 1.12 Axial loading of rockbolts at 2 m spacing ; cs_spacing = 2 round 5E-3 edge 1E-2 block 0,0 0,5 5,5 5,0 gen edge 1.0 group zone elastic zone model elastic density 1E3 bulk 1E9 shear 3E8 range group elastic boundary xvelocity 0 range -0.1, ,0.1 boundary yvelocity 0 range -0.1, ,0.1 boundary xvelocity 0 range -0.1, ,5.1 boundary xvelocity 0 range 4.9, ,5.1 struct rockbolt begin 2.5,5.0 end 2.5,2.5 seg 5 prop 1 struct prop 1 cs_scoh 1E20 cs_sstiff 1E9 e 1E10 density 4000 radius 1 & spacing 2 yield 1E20 ycomp 1E20 struct node 1 load 0.0, solve ratio 1.0E-5 save sp1.sav new ; divide properties by 2 round 5E-3 edge 1E-2 block 0,0 0,5 5,5 5,0 gen edge 1.0 group zone elastic zone model elastic density 1E3 bulk 1E9 shear 3E8 range group elastic

142 1-124 Special Features Structures/Fluid Flow/Thermal/Dynamics boundary xvelocity 0 range -0.1, ,0.1 boundary yvelocity 0 range -0.1, ,0.1 boundary xvelocity 0 range -0.1, ,5.1 boundary xvelocity 0 range 4.9, ,5.1 struct rockbolt begin 2.5,5.0 end 2.5,2.5 seg 5 prop 1 struct prop 1 cs_scoh 1E20 cs_sstiff 5E8 density 4E3 e 5E9 radius 1 yield & 1E20 ycomp 1E20 struct node 1 load 0.0, solve ratio 1.0E-5 save sp2.sav JOB TITLE : Spaced rockbolts UDEC (Version 5.00) LEGEND Oct :47:56 cycle time 8.171E+00 sec block plot Axial Force on Rockbolt Type # Max. Value rckblt E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.63 Actual axial forces in vertically loaded rockbolt at 2 m spacing (spacing given)

143 STRUCTURAL ELEMENTS JOB TITLE : Spaced rockbolts UDEC (Version 5.00) LEGEND Oct :48:13 cycle time 8.171E+00 sec block plot Axial Force on Rockbolt Type # Max. Value rckblt E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.64 Scaled axial forces in vertically loaded rockbolt at 2 m spacing (spacing not given) Symmetry Conditions Structural elements that lie on a line of symmetry should be assigned full properties for modulus and stiffness. Property values for cross-sectional area and yield strength should be reduced by 50% compared to the same property values for elements not on the symmetry line. Loads applied to structural elements on symmetry lines should also be reduced by 50% compared to the same loads applied away from the symmetry line Equilibrium Conditions The user must decide when the model has reached an equilibrium state. Equilibrium for problems involving structural elements can be determined by all the usual criteria (e.g., histories and velocity fields). However, if structural beam elements or rockbolts are used, an additional equilibrium criterion is available. At equilibrium, beam element or rockbolt element segments that share a common node will have equal and opposite moments. This can be confirmed with the PRINT struct element or PRINT rockbolt command. For certain types of structural-element problems (e.g., pull-tests on cables or rockbolts) a significant portion of the model region may develop nonzero components of velocity at the final state of solution. The default mechanical damping algorithm in UDEC can have difficulty damping this motion properly, because the mass-adjustment process requires velocity sign-changes (see Section in

144 1-126 Special Features Structures/Fluid Flow/Thermal/Dynamics Theory and Background). An alternative form of damping is available for this type of problem. This damping, known as combined damping or creep-type damping, is described in the Creep Material Models volume. Combined damping is invoked for the UDEC model with the DAMPING combined command Sign Convention Axial forces in all structural elements are positive in compression. Shear forces follow the opposite sign convention as that given for zone shear stresses (illustrated in Figure 2.46 in the User s Guide). Axial displacements for cable elements, rockbolt elements, beam elements and support members are positive for loading in compression. Axial displacements for reinforcement elements are positive for loading in tension. Normal forces at structural element interface contacts are positive in compression; normal displacements are positive for loading in tension. Moments at the end of beam and rockbolt elements are positive in the counterclockwise direction. Translational displacements at nodes are positive in the direction of the positive coordinate axes, and angular displacements are positive in the counterclockwise direction.

145 STRUCTURAL ELEMENTS References Azuar, J. J., et al. Le Renforcement des Massifs Rocheux par Armatures Passives (Rock Mass Reinforcement by Passive Rebars), in Proceedings of the 4th ISRM Congress (Montreux, September 1979), Vol. 1, pp Rotterdam: A. A. Balkema and The Swiss Society for Soil and Rock Mechanics (1979). Bjurstrom, S. Shear Strength on Hard Rock Joints Reinforced by Grouted Untensioned Bolts, in Proceedings of the 3rd International Congress on Rock Mechanics, Vol. II, Part B, pp Washington, D.C.: National Academy of Sciences (1974). Brierley, G. The Performance during Construction of the Liner of a Large, Shallow Underground Opening in Rock. Ph.D. Thesis, University of Illinois at Urbana-Champaign (1975). Coduto, D. P. Foundation Design: Principles and Practices. Prentice Hall (1994). CSIR MiningTEK. Personal communication (1993). Daemen, J. J. K. Tunnel Support Loading Caused by Rock Failure. Ph.D. Thesis, University of Minnesota; also available as U.S. Army Corps of Engineers Report MRD-3-75 (1975). Dight, P. M. Improvements to the Stability of Rock Walls in Open Pit Mines. Ph.D. Thesis, Monash University (1982). Dixon, J. D. Analysis of Tunnel Support Structure with Consideration of Support-Rock Interaction, U.S. Dept. of Interior, Bureau of Mines Investigation, Report RI7526 (June 1971). Donovan, K., W. E. Pariseau and M. Cepak. Finite Element Approach to Cable Bolting in Steeply Dipping VCR Stopes, in Geomechanics Applications in Underground Hardrock Mining, pp New York: Society of Mining Engineers (1984). Fuller, P. G., and R. H. T. Cox. Rock Reinforcement Design Based on Control of Joint Displacement A New Concept, in Proceedings of the 3rd Australian Tunnelling Conference (Sydney, Australia, 1978), pp Sydney: Inst. of Engrs., Australia (1978). Gerdeen, J. C., et al. Design Criteria for Roof Bolting Plans Using Fully Resin-Grouted Nontensioned Bolts to Reinforce Bedded Mine Roof, U.S. Bureau of Mines, OFR 46(4)-80 (1977). Haas, C. J. Shear Resistance of Rock Bolts, Trans. Soc. Min. Eng. AIME, 260(1), (1976). Hyett, A. J., W. F. Bawden and R. D. Reichert. The Effect of Rock Mass Confinement on the Bond Strength of Fully Grouted Cable Bolts, Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 29(5), (1992). Littlejohn, G. S., and D. A. Bruce. Rock Anchors State of the Art. Part I: Design, Ground Engineering, 8(3), (1975). Lorig, L. J. A Hybrid Computational Model for Excavation and Support Design in Jointed Media. Ph.D. Thesis, University of Minnesota (1984).

146 1-128 Special Features Structures/Fluid Flow/Thermal/Dynamics Lorig, L. J. A Simple Numerical Representation of Fully Bonded Passive Rock Reinforcement for Hard Rocks, Computers and Geotechnics, 1, (1985). Monsees, J. E. Station Design for the Washington Metro System, in Proceedings of the Engineering Foundation Conference Shotcrete Support, ACI Publication SP-54 (1977). Paul, S. L., et al. Design Recommendations for Concrete Tunnel Linings, University of Illinois, DOT Report No. DOT-TSC-UMTA (1983). Pells, P. J. N. The Behaviour of Fully Bonded Rock Bolts, in Proceedings of the 3rd International Congress on Rock Mechanics, Vol. 2, pp (1974). St. John, C. M., and D. E. Van Dillen. Rockbolts: A New Numerical Representation and Its Application in Tunnel Design, in Rock Mechanics Theory - Experiment - Practice (Proceedings of the 24th U.S. Symposium on Rock Mechanics, Texas A&M University, June 1983), pp New York: Association of Engineering Geologists (1983).

147 FLUID FLOW IN JOINTS FLUID FLOW IN JOINTS 2.1 Introduction UDEC has the capability to perform the analysis of fluid flow through the fractures of a system of impermeable blocks. A fully coupled mechanical-hydraulic analysis is performed, in which fracture conductivity is dependent on mechanical deformation and, conversely, joint fluid pressures affect the mechanical computations. The effects modeled in UDEC are summarized in Figure 2.1: Q Q Figure 2.1 Fluid/solid interaction in discontinua

148 2-2 Special Features Structures/Fluid Flow/Thermal/Dynamics This fluid-flow formulation is called compressible fluid, transient flow mode, and is the basic formulation for flow of a fluid in UDEC. Both confined flow and flow with a free surface can be modeled with this formulation. The fluid-flow calculation can also be run either coupled or uncoupled with the mechanical stress calculation. The formulation is described in Section In addition to the basic formulation, there are four other fluid-flow modes in UDEC: A computationally faster solution scheme than the basic formulation has been developed specifically for steady-state flow problems. This scheme, called the compressible fluid, steady state flow mode, is described in Section An alternative algorithm is available for transient flow, assuming an incompressible fluid. This formulation can greatly speed the calculation for transient analysis, but is restricted to simulations of the mechanical-fluid coupled response of systems undergoing quasi-static loading. This formation, called incompressible fluid, transient flow mode, is described in Section A special algorithm for modeling transient gas flow is also provided. The transient gas flow mode is described in Section Finally, two-phase liquid/gas flow in joints can be modeled. The transient two-phase fluid flow formulation is described in Section Special fluid-flow features are provided in UDEC. The joint permeability relation can be modified, and viscoplastic fluid flow can be simulated; this is described in Section A porous medium can be created around the UDEC block model to simulate a regional flow field. This algorithm is described in Section One-way thermal-hydraulic coupling of flow in joints in which temperature variations induce changes in the viscosity and density of water can also be simulated. This logic is discussed in Section Background information on the hydraulic behavior of rock joints and the associated fluid flow properties are described in Section 2.3. The UDEC commands required to perform a fluid-flow analysis are listed in Table 2.2, and discussed in detail in Section 2.4. Examples and verification problems illustrating each of the fluid-flow modes are given in Section 2.5.* * The data files in this section are stored in the directory ITASCA\UDEC500\Datafiles\Fluid with the extension.dat. A project file is also provided for each example. In order to run an example and compare the results to plots in this section, open a project file in the GIIC by clicking on the File / Open Project menu item and selecting the project file name (with extension.prj ). Click on the Project Options icon at the top of the Project Tree Record, select Rebuild unsaved states, and the example data file will be run, and plots created.

149 FLUID FLOW IN JOINTS Fluid-Flow Formulations Basic Algorithm Transient Flow of a Compressible Fluid The numerical implementation for fluid flow makes use of the domain structure described in Section 1 in Theory and Background (see Figure 2.2). For a closely packed system, there is a network of domains, each of which is assumed to be filled with fluid at uniform pressure and which communicates with its neighbors through contacts. Referring to Figure 2.2, domains are numbered 1 to 5: domains 1, 3 and 4 represent joints; domain 2 is located at the intersection of two joints; and domain 5 is a void space. As described in Section 1 in Theory and Background, domains are separated by the contact points (designated by letters A to F in the figure), which are the points at which the forces of mechanical interaction between blocks are applied. Because deformable blocks are discretized into a mesh of triangular elements, gridpoints may exist not only at the vertices of the block, but also along the edges. A contact point will be placed wherever a gridpoint meets an edge or a gridpoint of another block. For example, in the same figure, contact D implies the existence of a gridpoint along one of the edges in contact. As a consequence, the joint between the two blocks is represented by two domains: 3 and 4. If a finer internal mesh were adopted, the joint would be represented by a larger number of contiguous domains. Therefore, the degree of refinement of the numerical representation of the flow network is linked to the mechanical discretization adopted, and can be defined by the user. l D l E A B C D E F A... F... Contact Points Domains Figure 2.2 Flow in joints modeled as flow between domains In the absence of gravity, a uniform fluid pressure is assumed to exist within each domain. For problems with gravity, the pressure is assumed to vary linearly according to the hydrostatic gradient, and the domain pressure is defined as the value at the center of the domain. Flow is governed by the pressure differential between adjacent domains. The flow rate is calculated in two different ways, depending on the type of contact. For a point contact (i.e., corner-edge, as

150 2-4 Special Features Structures/Fluid Flow/Thermal/Dynamics contact F in Figure 2.2, or corner-corner), the flow rate from a domain with pressure p 1 to a domain with pressure p 2 is given by where k c = a point contact permeability factor, and q = k c p (2.1) p = p 2 p 1 + ρ w g(y 2 y 1 ) (2.2) where ρ w is the fluid density; g is the acceleration of gravity (assumed to act in the negative y-direction); and y 1,y 2 are the y-coordinates of the domain centers. In the case of an edge-edge contact, a contact length can be defined (e.g., in Figure 2.2, l D and l E denote the lengths of contacts D and E, respectively). The length is defined as half the distance to the nearest contact to the left plus half the distance to the nearest contact to the right. In this case, the cubic law for flow in a planar fracture (e.g., Witherspoon et al. 1980) can be used (see Section 2.3). The flow rate is then given by q = k j a 3 p l where k j is a joint permeability factor (whose theoretical value is 1/12μ); μ is the dynamic viscosity of the fluid; a is the contact hydraulic aperture; and l is the length assigned to the contact between the domains. (2.3) In UDEC, the user may change the permeability factor and exponent in Eq. (2.3). (See Section ) The above expression may be used for point contacts, provided a minimum length is assigned to these contacts. Eq. (2.2) indicates that flow may take place at a contact even when both domain pressures are zero; in this case, gravity may cause fluid to migrate from a domain that is not fully saturated. However, there are two factors to consider: (a) the apparent permeability should decrease as the saturation decreases in particular, permeability should be zero for zero saturation; and (b) fluid cannot be extracted from a domain of zero saturation. To address point (a), the flow rates (and, thereby, apparent permeability) computed by Eqs. (2.1) and (2.3) are multiplied by a factor f s, a function of saturation, s, f s = s 2 (3 2s) (2.4)

151 FLUID FLOW IN JOINTS 2-5 The function is empirical but has the property that f s = 0ifs = 0, and f s = 1ifs = 1 (i.e., permeability is unchanged for full saturation, and zero for zero saturation). Further, the derivative of Eq. (2.4) is zero at s = 0 and s = 1, which is reasonable to expect, physically. The value of s used in Eq. (2.4) is taken as the saturation of the domain from which inflow takes place; hence, inflow cannot occur from a completely unsaturated domain. The hydraulic aperture is given, in general, by a = a o + u n (2.5) where a o is the joint aperture at zero normal stress; and u n is the joint normal displacement (positive denoting opening). A minimum value, a res, is assumed for the aperture, below which mechanical closure does not affect the contact permeability. A maximum value, a max, is also assumed, for efficiency, in the explicit calculation (arbitrarily set to five times a res, but it can be changed by the user). The variation of aperture with normal stress on the joint is depicted in Figure 2.3. The above expression is a very simple relation between joint mechanical and hydraulic apertures; more elaborate relations, such as the empirical law proposed by Barton et al. (1985), might also be used. a a max a 0 a res (compressive) σ n Figure 2.3 Relation between hydraulic aperture, a, and joint normal stress, σ n, in UDEC At each timestep in the mechanical calculation in UDEC, the computations determine the updated geometry of the system, thus yielding the new values of apertures for all contacts and volumes of all domains. Flow rates through the contacts can be calculated from the above formulas. Then, domain

152 2-6 Special Features Structures/Fluid Flow/Thermal/Dynamics pressures are updated, taking into account the net flow into the domain, and possible changes in domain volume due to the incremental motion of the surrounding blocks. The new domain pressure becomes p = p o + K w Q t V K V w V m (2.6) where p o is the domain pressure in the preceding timestep; Q is the sum of flow rates into the domain from all surrounding contacts; K w is the bulk modulus of the fluid; V = V V o ; and V m = (V + V o )/2, where V and V o are the new and old domain areas, respectively. If the new domain pressure computed by Eq. (2.6) is negative, then the pressure is set to zero, and the domain outflow is used to reduce saturation, s, as follows. s = s o + Q t V V V m (2.7) where s o is the domain saturation at the preceding timestep. The pressure remains at zero as long as s<1; in this case, Eq. (2.7) is applied instead of Eq. (2.6). If the computed s is greater than 1, then s is set to 1, and Eq. (2.6) is used again. This scheme ensures that fluid mass is conserved; the excess domain volume is either used to change pressure or to change saturation. The phreatic surface (boundary between nodes with s = 1, and nodes of s<1) arises naturally from the algorithms described above. Given the new domain pressures, the forces exerted by the fluid on the edges of the surrounding blocks can be obtained. These forces are then added to the other forces to be applied to the block gridpoints, such as the mechanical contact forces and external loads. As a consequence of this procedure, total stresses will result inside the impermeable blocks, and effective normal stresses will be obtained for the mechanical contacts. Numerical stability of the present explicit fluid flow algorithm requires that the timestep be limited to t f [ = min V K w i k i ] (2.8) where V is domain volume, and the summation of permeability factors, k i, is extended to all contacts surrounding the domain (i.e., k i = max(k c,k j a 3 /l). The minimum value of t over all domains is used in the analysis.

153 FLUID FLOW IN JOINTS 2-7 For transient flow analysis, the numerical stability requirements may be rather severe, and may make some analyses very time-consuming or impractical, especially if large contact apertures and very small domain areas are present. Furthermore, the fluid filling a joint also increases the apparent joint stiffness by K w /a, thus possibly requiring a reduction of the timestep used in the mechanical calculation Steady-State Flow Algorithm In many studies, only the final steady-state condition is of interest. In this case, several simplifications that make the present algorithm very efficient for many practical problems are possible. The steady-state condition does not involve the domain volumes. Thus, these can be scaled to improve the convergence to the solution. A scheme that was found to produce good results consists of assigning to a given domain a volume V that, inserted in the timestep expression above, leads to the same timestep for all domains. The contribution of the change in domain volume to the pressure variation can also be neglected, thus eliminating the influence of the fluid stiffness in the mechanical timestep, and making it unnecessary to specify fluid bulk modulus. Furthermore, as the steady-state condition is approached, the pressure variation in each fluid step becomes very small, allowing the execution of several fluid steps for each mechanical step without loss of accuracy. An adaptive procedure that triggers the update of the mechanical quantities whenever the maximum increment of pressure in any domain exceeds some prescribed tolerance (for example, 1% of the maximum pressure) was implemented in UDEC Transient Flow of an Incompressible Fluid In the standard formulation described in Section 2.2.1, the fluid pressure increments are calculated from the joint volume variation and the net inflow into the domain (Eq. (2.6)). For small joint apertures, the fluid appears to be a stiff spring, with a stiffness higher than the typical joint stiffness. In an explicit algorithm, this implies that the mechanical timestep must be reduced. The fluid timestep, calculated by Eq. (2.8), is inversely proportional to the bulk modulus and joint conductivity. For typical joint apertures, fluid timesteps on the order of milliseconds are obtained. Therefore, this algorithm can only be applied in practical problem settings to short duration simulations. A different procedure, that overcomes the difficulties described above, was developed. Before presenting this scheme, it is worthwhile to review the essential characteristics of a fluid-rock system and the particular conditions to be modeled. The characteristics of a rock-fluid system can be summarized as follows. 1. There are two distinct difficulties that modelers confront: a. the fluid trapped in a joint appears to be very stiff, owing to the small aperture; and b. permeability varies rapidly with changing aperture, owing to the cubic term in the flow equation.

154 2-8 Special Features Structures/Fluid Flow/Thermal/Dynamics The two difficulties are separate and can be addressed separately. For example, if the imposed pressure changes are small compared to the existing pressures (and rock stresses), then item (b) is unimportant. 2. It is the rock, rather than the fluid, that determines fluid pressure. In a conventional pipe network, for example, the fluid determines its own pressure, via the flow and continuity equations. However, a typical rock block is so soft compared to the fluid trapped in a joint (factors of 10 3 to 10 4 are common) that significant changes in fluid volume hardly affect rock stresses. Since the rock s normal stresses must balance the fluid pressure in the neighboring joints, the fluid pressure is determined by the rock stress. It then follows that spatial variations in rock stress are directly responsible for the direction and magnitude of flow since flow occurs in response to pressure gradients. 3. If the interest is in nonsteady flow but not dynamic flow (i.e., the model must accurately capture the transmission delay as pressure fluctuations migrate from one part of the system to another), then inertial effects or wave-propagation effects can be neglected. We confine our attention to a liquid, such as water, contained in joints of small aperture (in the range 10 to 100 micrometers) and length in the range1mto10m. 4. Time constants (e.g., time to reach a given fraction of the final pressure or time for a pressure change to propagate a certain distance) depend more on the compliance of the whole system and the mode of deformation than on local compliance; the bulk modulus of the fluid is almost irrelevant. As an extreme example, imagine fluid injected into a series of joints bounded by blocks loaded by deadweight only (see Figure 2.4). The speed of propagation in this case is zero, since no pressure gradients develop between blocks. As stiffness is added between blocks, the propagation speed increases. Keeping these points in mind, the following scheme is proposed. Flow rate is calculated from pressure difference in the usual way, as expressed previously in Eq. (2.3). At each domain (intersection of several joints or the middle part of a joint), the flow contributed by each joint is added algebraically and multiplied by the fluid timestep to obtain the net fluid volume entering the domain: V f = q t f (2.9) Instead of trying to translate this volume immediately into rock displacements, we imagine that the excess fluid is stored in a balloon attached to the domain. We then hold flow time constant, and allow the contents of each balloon to leak into its associated domain. This leakage stops when the increase in domain volume becomes equal to the volume stored in the balloon. The process involves the usual dynamic relaxation of the equations of motion of the gridpoints, but with an additional pressure boundary condition supplied by the leakage of fluid into the domain. The following leakage scheme was found to be satisfactory.

155 FLUID FLOW IN JOINTS 2-9 p = p 0 + F p ( V stored V domain ) (2.10) where p and p 0 are the domain pressures at the new and old (mechanical) timesteps, respectively, V stored is the volume originally stored in the balloon, V domain is the volume increase of the domain, and F p is a constant factor. Eq. (2.10) can be viewed as a kind of servo-control that adjusts pressure until the domain volume has increased by exactly the volume stored in the balloon. There are no problems of compatibility, as we are simply applying a pressure boundary condition, rather than enforcing a velocity condition. During the leakage phase, we are solving all the mechanical equations by dynamic relaxation, so condition 4 (given above) is satisfied the fluid sees the compliance of the entire system. (dead loads) W W W W W W Q Figure 2.4 Extreme example in which the speed of propagation depends on system stiffness The algorithm proceeds by performing a sequence of fluid steps, the timestep being defined by the user. For each step, a series of mechanical relaxation steps is performed in order to achieve continuity of flow at each domain. Given the assumption of fluid incompressibility, the net flow into a domain during a fluid step must equal the increment of domain volume. The unbalanced fluid volume, being the difference between the two, is gradually reduced during the relaxation procedure. For this purpose, the domain pressure is increased or reduced proportionally to the unbalanced volume for each domain. The proportionality factor is controlled by an adaptive scheme, and therefore varies during the iterations to provide better convergence.

156 2-10 Special Features Structures/Fluid Flow/Thermal/Dynamics Transient Flow of a Compressible Gas Transient flow of a compressible gas through joints can also be simulated in UDEC. The flow algorithm is based upon the algorithm used to model compressible flow of a fluid (see Section 2.2.1), with the following modifications. Modeling the flow of a gas requires consideration of the strong density dependence on pressure, which can usually be neglected when dealing with slightly compressible fluids, as is done with the other flow modes in UDEC. For an ideal gas, we can consider the flow under isothermal conditions, for which and adiabatic flow, for which ρ g = Bp (2.11) ρ g = Bp 1/γ (2.12) where ρ g is gas density, p is pressure, and B and γ are constants (see Chan et al. 1993). As a consequence, the gas bulk modulus is also a function of pressure. In the isothermal case, we have, from Eq. (2.11), the bulk modulus given by For the adiabatic case, gas bulk modulus is K g = p (2.13) K g = p γ (2.14) The dependence of density on pressure requires that the spatial variation of density be taken into account in the fluid mass balance (nρ g ) t + (ρ gq i ) x i = 0 (2.15) In contrast with the existing logic in UDEC for slightly compressible fluids, the analysis of gas flow requires the consideration of the convective terms, resulting from the density gradients in the second term of Eq. (2.15). Currently, for water flow in UDEC, the domain pressures are updated at every step via Eq. (2.6), which can be written as

157 FLUID FLOW IN JOINTS 2-11 p new = p old + K g 1 V i q i t K g V V (2.16) where the summation extends over all the contacts that allow flow into or out of the domain. For gas flow, the density varies from domain to domain. As flow rates are calculated at the contacts that separate the domains, it is necessary to assign a density to the flow in and out of each domain. Assuming that the fluid density at a contact is given by the average of the domain fluid densities, we have, at contact i, ρ im = ρ i1 + ρ i2 2 (2.17) where ρ i1 and ρ i2 are the densities of the two domains. The mass balance of the flow in and out of the domain implies that the second term on the right-hand side of Eq. (2.16) needs to be adjusted by a factor equal to the ratio of the density ρ im at the contact and the old domain density, ρ. p new = p old + K g 1 V i ρ im q i ρ t K V g V (2.18) Density and bulk modulus for each domain are updated as a function of domain pressure, according to Eqs. (2.11) to (2.14) Two-Phase Flow in Joints Two-phase flow, fully coupled with deformation of solid blocks, can be simulated in UDEC. The flow is considered to take place in the joints only. The blocks are assumed to be completely impermeable. Both phases are active in this formulation (i.e., UDEC calculates pressure changes and flow for both wetting and non-wetting fluid). Capillary effects can be included. The capillary pressure curve and relative permeability of phases (as a function of saturation) have a predefined functional form, with parameters that can be set by the user. The two-phase logic is based upon the default, compressible transient flow logic, as described in Section The default flow logic in UDEC can simulate (in two dimensions) flow of a compressible fluid through fractures in a deformable rock mass. The code can analyze both confined and unconfined flows. Modeling of unconfined flow requires logic for determining the phreatic surface as a function of the saturation of rock joints by a wetting phase (e.g., the water is the wetting phase in a contact between the water and the air). Thus, a simple two-phase flow can be simulated with the default fluid-flow logic in UDEC. However, the assumption of this model is that the air (the non-wetting phase) is inactive: there is no movement of the air since the pressure in the air is uniform (under atmospheric conditions).

158 2-12 Special Features Structures/Fluid Flow/Thermal/Dynamics The two-phase flow logic allows for independent specification of general pressure and saturation boundary and initial conditions for each of the fluid phases. The algorithm does not rely on any assumptions about the number or location of interfaces between phases. The shape, the number and the connectivity of the interfaces between two phases are completely general. The length scale in the calculation is the length of the joint in the model. Rock joint roughness is not considered explicitly, but is taken into account by constitutive relations imbedded into the relations between (1) flow rates and pressure gradient, (2) the capillary pressure and the saturation and joint opening, and (3) the phase-relative permeability and the saturation. The following equations govern transient two-phase flow of immiscible, compressible fluids in a network of one-dimensional joints in a deformable, discontinuous solid. s w + s nw = 1 p nw p w = p c (s w,a) (ρ wq w ) x (ρ nwq nw ) x q w = κ w (s w ) p w x q nw = κ nw (s nw ) p nw = ρ w ( sw t = ρ nw ( snw t x + s w a + s nw a da dt + s w p w K w t ) da dt + s nw p nw K nw t ) (2.19) where subscripts w and nw denote wetting phase (e.g., water) and non-wetting phase (e.g., gas), respectively, p is the pressure, p c is the capillary pressure, s is the saturation, κ is the joint permeability, q is the flow rate, a is the joint aperture, and K is the bulk modulus. Note that the bulk modulus of both the wetting and non-wetting fluid is considered to be constant (i.e., independent of pressure). Both fluids are considered to be slightly compressible the spatial gradients of the fluid density are neglected. In the model defined by Eq. (2.19), the interface between two phases is a region of localized gradient of saturation. For the purpose of interpretation of model results, the interface can be defined as a set of points in the model plane where saturation is equal to 0.5, although the transition region (of localized saturation gradient) may be spread over a number of joint segments. Flow of each phase is governed by the pressure differential between adjacent domains. As in the single-phase flow, the flow rate is calculated in two different ways, depending on the type of contact. For a point contact, the flow rates are given by q w = f w (s w )k wc p w q nw = f nw (s nw )k nwc p nw (2.20)

159 FLUID FLOW IN JOINTS 2-13 where f w (s) is a wetting fluid relative permeability factor; f nw (s) is a non-wetting fluid relative permeability factor; k wc is a wetting fluid point contact permeability factor; is a non-wetting fluid point contact permeability factor; and k nwc p w = p w2 p w1 + ρ w g(y 2 y 1 ) p nw = p nw2 p nw1 + ρ nw g(y 2 y 1 ) (2.21) where ρ w is the wetting fluid density; ρ nw is the non-wetting fluid density; p w1 and p w2 are wetting fluid pressures in the domains 1 and 2; p nw1 and p nw2 are non-wetting fluid pressures in the domains 1 and 2; g is the acceleration of gravity (assumed to act in the negative y-direction); and y 1,y 2 are the y-coordinates of the domain centers. The flow rates along the planar fracture are given by q w = f w (s w )k wj a 3 p w l q nw = f nw (s nw )k nwj a 3 p nw l where k wj is a wetting fluid joint permeability factor whose theoretical value is (1 / 12μ w ); μ w is the dynamic viscosity of the wetting fluid; k nwj is a non-wetting fluid joint permeability factor with theoretical value (1 / 12μ nw ); μ nw is the dynamic viscosity of the non-wetting fluid; a is the contact hydraulic aperture; and l is the length assigned to the contact between the domains. (2.22) The two factors f w and f nw account for the effect of reduction in conductivity of either phase in the case when a joint is partially saturated. The functional form of the two factors and their dependence on joint properties (e.g., joint roughness) is not well known. It has been assumed in the current formulation that the two relative permeability factors depend on joint saturation only, as it is given by the relations f w = sw 2 (3 2s w) f nw = snw 2 (3 2s nw) (2.23)

160 2-14 Special Features Structures/Fluid Flow/Thermal/Dynamics Capillary pressure curves (i.e., dependence of capillary pressure to saturation) is relatively well known for soils. The standard laboratory experiments are used for measurements of capillary pressure curves. Various authors have proposed analytical expressions for describing general shapes of capillary pressure curves. The capillary pressure curves for rock joints have not been investigated as extensively as curves for soils. Pruess and Tsang (1990) analytically derived a relation between the average joint apertures, joint roughness, saturation and the capillary pressure. Unfortunately, that curve was not convenient for implementation in UDEC. The curve of the following form was implemented in UDEC as a capillary pressure curve. p c (s w,a)= γρ w g a 0 a ( β 1/β 1 β sw 1) (2.24) where a 0, β and γ are curve parameters. a 0 is the reference hydraulic aperture (dimension of length) for which parameters γ and β are determined. (The capillary pressure is inversely proportional to hydraulic aperture, a.); β controls the shape of the curve (as illustrated in Figure 2.5). γ has dimension of length and scales the capillary pressure curve with density of wetting fluid (i.e., defines the height of capillary fringe). 1E+15 1E+14 1E+13 1E+12 1E+11 1E+10 1E+09 beta = 1.1 beta = 2 beta = 10 1E+08 p c /γ ρw g 1E+07 1E s w Figure 2.5 Capillary pressure curve as a function of saturation and parameter β

161 FLUID FLOW IN JOINTS 2-15 Eq. (2.24) has the same form as the van Genuchten curve, which is very often used for soils. The difference is that the curve defined by Eq. (2.24) takes into account the effect of hydraulic aperture. Note that the capillary pressure curve becomes singular for zero saturation. Simulation of the models with very small saturation would impose a very severe restriction on the critical calculation timestep. Therefore, the model has the lower bound on saturation defined by SET satmin. (The default value for satmin is 10 4.) The domain values of a 0, β and γ are specified by the command PROPERTY dmat n acap, bcap and gcap. The domain wetting fluid pressure changes (due to flow only) can be expressed, after solving a system of four difference equations (obtained from Eq. (2.19)): p w = K p w K nw (Q w + Q nw ) K w S c nw s w Q w t p K w s nw + K nw s w s nw s c w V s w (2.25) The change in saturation is s w = K w s nw Q w K nw S w Q nw K w s nw + K nw s w s nw s w p c t V s w (2.26) Similarly, the change in the domain wetting fluid pressure (due to joint deformation only) is p w = K w [ V ( p c s w s w s nw K nw ) s nw p c a a ] K w s nw + K nw s w s nw s w p c s w (2.27) and the change in saturation is s w = s w s nw [ V (K w K nw ) + p c a a ] K w s nw + K nw s w s nw s w p c s w (2.28) Therefore, the pressures and saturation in the domain in the next timestep are p w = p w0 + p w s w = s w0 + s w s nw = 1 s w (2.29) p nw = p c (s w ) + p w

162 2-16 Special Features Structures/Fluid Flow/Thermal/Dynamics where p w0 is the domain wetting fluid pressure in the current timestep; p w is the domain wetting fluid pressure in the next timestep; s w0 is the domain saturation in the current timestep; s w is the domain saturation in the next timestep; s nw = 1 s w ; p nw is the domain non-wetting fluid pressure in the next timestep; Q w is the sum of wetting fluid flow rates into the domain from all surrounding contacts; Q nw is the sum of non-wetting fluid flow rates into the domain from all surrounding contacts; K w is the bulk modulus of the wetting fluid; K nw is the bulk modulus of the non-wetting fluid; V = V V o increment in domain volume between current and next timestep; a = a a o increment between the current and the next timestep in the average aperture of the joints that form the domain; p c s w partial derivative of the capillary pressure curve with the respect to saturation; and p c a partial derivative of the capillary pressure curve with the respect to average aperture. Given the new domain pressures, the average pressure exerted by the fluids on the edges of the surrounding blocks can be obtained: p av = p w s w + p nw s nw If two-phase flow is on (SET flow twophase), then pressures of both wetting and non-wetting fluid change as a function of domain volumetric deformation (as given by Eq. (2.27)). Numerical stability of the present explicit fluid flow algorithm requires that the timestep be limited to t f [ = min V V K 1 i k, 1i K 2 i k 2i ] (2.30) where V is domain volume, and the summation of permeability factors is extended to all contacts surrounding the domain. The symbols in Eq. (2.30) are defined:

163 FLUID FLOW IN JOINTS 2-17 k 1i = max(k wc,k nwc,k wj a 3 /l,k nwj a 3 /l) K 1 = max(k w,k nw ) k 2i = max(k wc,k wj a 3 /l) K 2 = max(k w, p c s w ) (2.31) The minimum value of t over all domains is used in the analysis. For transient flow analysis, the numerical stability requirements may be rather severe and may make some analyses very time-consuming or impractical, especially in cases when wetting fluid with large bulk modulus and non-wetting fluid with small viscosity (i.e., large conductivity of non-wetting fluid) are both in the same model Modifying Joint Flow Rate and Simulating Viscoplastic Flow in Joints The flow of a Bingham body (or liquid), such as cement grout, is of the viscoplastic type. The major difference between this model and that for a Newtonian liquid is that, for a Bingham fluid, a yield stress, τ y, must be exceeded to initiate flow. For Newtonian flow, it is assumed that the flow rate per unit width, q, is related linearly to the pressure gradient, J, as shown in Figure 2.6. The general equation for fluid flow between planar surfaces is given by q = bax J 12μ (2.32) where a = fracture width (aperture); b = empirical coefficient; μ = dynamic viscosity of fluid; and x = aperture exponent. In the most widely used form of this relation, known as the cubic flow law, x = 3 and b = 1 (see Eq. (2.3)). The values for x and b can be specified by the user with the JOINT property keywords expa and empb, respectively. The flow gradient relation of a Bingham body is similar to this expression, except that no flow occurs until the threshold gradient, J 0, is exceeded, as shown in Figure 2.7.

164 2-18 Special Features Structures/Fluid Flow/Thermal/Dynamics q ba x 12 μ J Figure 2.6 Flow-gradient relation for Newtonian fluid in UDEC q ba x 12 μ J 0 J Figure 2.7 Flow-gradient relation for Bingham fluid in UDEC

165 FLUID FLOW IN JOINTS 2-19 Considering the balance of forces on a rectangular element of fluid, the expression for the threshold gradient for flow between parallel sides of aperture, a, isgivenby J = 2 τ y a (2.33) where τ y is the yield stress. The expression for the threshold gradient can also be obtained by considering the equation for steady laminar flow of a Bingham plastic in a circular pipe. This equation is known as Buckingham s equation (Wilkinson 1960): Q = πr4 P 8Lμ p [ [ 2L τ y r P ] 1 [ 2L τ y + 3 r P ] 4 ] (2.34) where Q = volume rate of flow; r = pipe radius; μ p = Bingham plastic viscosity; and P /L = pressure gradient = J. From this expression, it can be seen that no flow occurs if the pressure gradient, J, is zero or equals 2 τ y /r. It is not clear from the equation what occurs at pressure gradients between zero and 2 τ y /r, but it is reasonable to assume that no steady flow occurs within this range. Therefore, the threshold gradient, J 0, the gradient at which steady flow is possible, is given by J 0 = 2 τ y r (2.35) Note that this expression can also be derived by considering the balance of forces acting on a cylindrical element of fluid with radius r and length L. For an example application of this feature, see Section 5 in the Example Applications. This example applies the viscoplastic flow model to simulate cement grouting. The threshold pressure gradient, J 0, is specified with the FLUID cohw command.

166 2-20 Special Features Structures/Fluid Flow/Thermal/Dynamics Fluid Boundary Logic A porous medium may be wrapped around the UDEC block model in order to simulate fluid flow on a larger scale while keeping the number of blocks reasonable. A radial mesh is created around the outer boundary of UDEC. The outer boundary of the mesh is circular and has imposed fluid pressures. The inner boundary is the outer boundary of the UDEC block model. The variation of pressures across this inner boundary is continuous, and flow to or from a joint is accounted for in the grid zone next to it in order to satisfy the fluid-mass balance. An example mesh is given in Figure 2.8: Figure 2.8 Porous medium mesh The formulation is described below. The logic, at present, is only intended for steady-state flow; a relaxation factor is used for convergence. The formulation assumes small-strain conditions and confined flow; a free surface cannot be modeled. Darcy s law for an anisotropic porous medium is V i = K ij p x j where V i is the specific discharge vector, p is the pressure and K ij is the permeability tensor. Each quadrilateral element in the porous medium mesh is divided into triangles in two different ways (see Figure 2.9(a)). The specific discharge vector can be derived for the generic triangle of Figure 2.9(b). By Gauss theorem,

167 FLUID FLOW IN JOINTS 2-21 Hence, p = 1 pn i ds (2.36) x i A s V i K ij A pnj s (2.37) where is the summation over the three sides of the triangle. For the x-component of V i, V 1 = 1 A [ ] K 11 pn1 s + K 12 pn2 s (2.38) The contribution of side (ab) of the triangle to the summation is V (ab) 1 = 1 [ ] K 11 (p (b) + p (a) )(x (b) 2A 2 x (a) 2 ) + K 12(p (b) + p (a) )(x (b) 1 x (a) 1 ) (2.39) Similarly, V (ab) 2 = 1 [ ] K 21 (p (b) + p (a) )(x (b) 2A 2 x (a) 2 ) + K 22(p (b) + p (a) )(x (b) 1 x (a) 1 ) (2.40) The other two sides, (bc) and (ca), provide similar contributions to V i. This specific discharge vector is then converted to scalar volumetric flow rates at the nodes by making dot products with the normals to the three sides of the triangle. The general expression is q = V in i s 2 (2.41) where the factor of 2 accounts for the fact that we take the average of the contribution from the two triangle-pairs that make up the quadrilateral element. The flow rate into node (a) in terms of coordinates is then q (a) = 1 2 { } V 1 (x (b) 2 x (c) 2 ) + V 2(x (b) 1 x (c) 1 ) (2.42)

168 2-22 Special Features Structures/Fluid Flow/Thermal/Dynamics A B b s + a D C c (a) (b) Figure 2.9 Elements in porous medium mesh Similar expressions apply to nodes (b) and (c). Nodal flow rates are added from the other three triangles shown in Figure 2.9(a). The stiffness matrix [K] of the whole quadrilateral element is defined in terms of the relation between the pressures at the four nodes and the four nodal flow-rates, as derived above: {q} =[K]{p} (2.43) For elements along the inner boundary, contributions from the corresponding joints are added to these flow rates. An example application of the fluid boundary is given in Section One-Way Thermal-Hydraulic Coupling Temperature variations induce changes in the viscosity and density of water. Because water viscosity governs the conductivity of joints, and water density governs the magnitude of hydraulic pressure gradients, these effects should be taken into account when flow is simulated in a nonhomogeneous or time-varying temperature field. Note that a full thermal-hydraulic coupling would require the fluid flow to influence heat transfer. This behavior is not included in this version of UDEC.* The thermal logic assumes that the rock mass is continuous, and does not account for water flow in the joints. An example application of the thermal coupling is given in Section * Other researchers have adapted UDEC for simulation of coupled thermal convection of fluid in fractures (e.g., see Abdallah et al. 1995).

169 FLUID FLOW IN JOINTS Hydraulic Behavior of Rock Joints Parallel Plate Model Flow in planar rock fractures may be idealized by means of the parallel plate model. The analytic solution for laminar viscous flow between parallel plates gives the mean velocity as v = k f J (2.44) where J is the hydraulic gradient, and the fracture hydraulic conductivity is given by k f = a2 g 12ν where a is the fracture width; ν is the kinematic viscosity of the fluid; and g is the acceleration of gravity. (2.45) The flow rate per unit width can thus be expressed as which is usually referred as the cubic flow law. q = va= a3 g 12ν J (2.46) Since pressure (p) is equal to gρ w h (where h is the head), and dynamic viscosity (μ) is equal to νρ w, then Eq. (2.44) can also be written as (i.e., see Eq. (2.3)). q = a3 12μ Experiments conducted by Louis (1969) showed that this law is essentially valid for laminar flow in rock joints. This author proposed an empirical correction factor for the above expression in order to account for fracture roughness. Witherspoon et al. (1980) tested both open and closed joints. They reported that the cubic law is still valid for the latter, provided that the actual mechanical aperture is used. Due to the effects of roughness and tortuosity of flow, the fracture conductivity in their experiments was reduced by a factor between 1.04 and Barton et al. (1985) proposed an empirical formula that gives the hydraulic aperture (to be used in the cubic law) as a function of the mechanical aperture and the joint roughness coefficient (JRC). p l

170 2-24 Special Features Structures/Fluid Flow/Thermal/Dynamics Fluid Flow Properties and Units Typical SI units for the various parameters described in Sections 2.2 and 2.3 are given in Table 2.1: Table 2.1 Parameter q p kc k j a l Q Kw Typical SI units for fluid flow parameters SI Units m 3 /sec m Pa m 2 /sec Pa 1/(Pa sec) m m m 3 /sec Pa An example calculation is given for fluid flow parameters to use in a UDEC model. The following discussion relates primarily to the gravity dam example problem presented in Section 4 in the Example Applications, but it can serve as a starting point for other problems as well. In the example, joint hydraulic apertures were calculated from Eq. (2.5). A residual aperture, a res, was taken as the minimum hydraulic aperture. A maximum value for the hydraulic aperture equal to twice the residual aperture was assumed for reasons of computational efficiency, because the timestep required for stability of the fluid flow algorithm is inversely proportional to joint conductivity. As conductivity is proportional to the cube of the aperture, considerable variation of permeability due to stress changes can still be modeled despite this constraint. In the base run, the values a 0 =1mmandares = 0.5 mm were used. The water properties used were ρ w = 1000 kg/m 3 and μ = Pa sec. (This problem is a steady-state flow analysis, so K w is not required as input, but is automatically defined, as discussed in Section ) The dynamic viscosity of water (μ = 10 3 Pa sec) was used to calculate the joint permeability factor in UDEC (see Eq. (2.3)): k j = 1 12μ = 83.3 Pa 1 sec 1 (2.47) See the data file for the gravity dam example at the end of Section 4 in the Example Applications.

171 FLUID FLOW IN JOINTS Calculation Modes and Commands for Fluid-Flow Analysis A summary of the different fluid-flow calculation modes and a brief review of commands related to fluid flow analysis are presented in this section. The commands are summarized by function in Table 2.2: Table 2.2 Summary of fluid flow commands Compressible Liquid, Compressible Liquid, Incompressible Liquid, Compressible Gas, Two-Phase Fluid, Steady-State Flow Transient Flow Transient Flow Transient Flow Transient Flow set flow SET flow steady SET flow compressible SET flow incompressible SET flow gas SET flow twophase mode 1 joint flow JOINT jperm JOINT jperm JOINT jperm JOINT jperm JOINT jperm properties azero ares azero ares azero ares azero ares azero ares expa 4 empb 4 expa 4 empb 4 expa 4 empb 4 expa 4 empb 4 expa 4 empb 4 nwjperm SET capratio j5flow SET capratio j5flow SET capratio j5flow SET capratio j5flow SET capratio j5flow fluid FLUID density 2 dtable 3 FLUID bulkw FLUID density 2 dtable 3 FLUID density 2 FLUID bulkw properties cohw 4 density 2 dtable 3 cohw 4 gas_alpha density 2 dtable 3 ktable 3 cohw 4 ktable 3 gas_constant cohw 4 ktable 3 gas_densitymin nwdens 2 gas_bulkmin nwbulk ktable 3 ktable 3 domain SET capmin SET capmin SET capmin PROP acap bcap gcap properties SET capmin satmin satmax specify fluid flow boundary conditions 5 BOUND pp pxgrad pygrad impermeable BOUND pp pxgrad pygrad impermeable BOUND pp pxgrad pygrad impermeable BOUND pp pxgrad pygrad impermeable BOUND pp pxgrad pygrad impermeable nwpp nwpx nwpy nwimperm saturation satxgraad satygrad seepage initialize fluid flow time for boundary histories FBOUND SET reftime=flow ftime SET reftime=flow ftime SET reftime=flow ftime INSITU pp pxgrad pygrad INSITU pp pxgrad pygrad INSITU pp pxgrad pygrad INSITU pp pxgrad pygrad INSITU initialize fluid ywtable ywtable ywtable ywtable pressure in PFIX PFIX PFIX PFIX PFIX domains PFREE PFREE PFREE PFREE PFREE WELL WELL WELL WELL WELL initialize fluid INSITU zone_pp INSITU zone_pp INSITU zone_pp INSITU zone_pp INSITU pressure in blocks ZONE biot_coef ZONE biot_coef ZONE biot_coef ZONE biot_coef ZONE SET reftime=flow ftime pp pxgrad pygrad ywtable zone_pp biot_coef flow-only SET mech=off SET mech=off SET mech=off SET mech=off solution CYCLE n 7 CYCLE n 6 CYCLE n 6 CYCLE n 6 coupled SET ptol SET nfmech nflow SET dtflow SET nfmech nflow SET nfmech nflow mechanical-flow FRACTION fb fz fw 8 voltol maxmech solution FRACTION fb fz fw 8 FRACTION fb fz fw 8 CYCLE n 7 CYCLE n 9 CYCLE n 6 CYCLE n 9 CYCLE n 9 SOLVE time ftime time time output options PRINT domain contact flow max joint PRINT domain contact flow max joint PRINT domain contact flow max joint PRINT domain contact flow max joint PRINT domain contact flow max joint PLOT pp flow vflow PLOT pp flow vflow PLOT pp flow vflow PLOT pp flow vflow PLOT pp flow vflow aperture aperture aperture aperture aperture joint jline joint jline joint jline joint jline joint jline HIST flowrate pp HIST flowrate pp HIST flowrate pp HIST flowrate pp HIST flowrate pp flowtime flowtime unbvol flowtime flowtime 1 Fluid flow can be turned on and off by using the commands SET flow on and SET flow off 2 Use only if gravity is acting. 3 Use only for temperature dependency. 4 Use to modify permeability relation or for visco-plastic flow 5 Mechanical-stress boundary conditions must be specified separately from fluid flow conditions 6 n =fluid flow steps 7 n =calculation steps to reach steady-state flow 8 Use to adjust Dt f =Dt z or Dt f =Dt b for dynamic analysis 9 n =mechanical steps

172 2-26 Special Features Structures/Fluid Flow/Thermal/Dynamics Selection of Calculation Mode Five calculation modes are available in UDEC for different types of fluid-flow problems. The fluid flow modes are accessed by first setting the fluid-flow configuration in UDEC by specifying the CONFIG fluid command. (1) Transient Flow Analysis Compressible Fluid (SET flow compressible) This option is selected by the SET flow compressible command and is the basic formulation, as described in Section The fluid bulk modulus given in the FLUID command is used in the calculation of the transient response of the system. The explicit nature of the algorithm requires the fluid timestep to be limited to ensure numerical stability. For typical ranges of properties, the timesteps are very small in relation to the flow time scales, and this algorithm is only practical for simple problems or short flow times. The main application of compressible fluid analysis is in the modeling of the pressure response of fluid-filled joints to a short-duration dynamic loading (e.g., an earthquake). For these short time scales, actual flow does not change significantly, and the flow calculation may be turned off. However, if the fluid bulk modulus is defined, the fluid pressure will still vary due to joint deformation, and may induce slip or separation. Note that the apparent stiffness of the fluid (given by K w /a, the ratio of the fluid bulk modulus to the joint aperture) will reduce the mechanical timestep, so very small residual apertures may lead to large runtimes. (2) Steady-State Flow Analysis (SET flow steady) In many applications, only the steady-state regime is of interest. For these cases, a special algorithm that provides a fast convergence to the steady-state solution (see Section 2.2.2) is available in UDEC. This option is selected with the SET flow steady command. The algorithm is based on the scaling of the fluid bulk modulus and of the domain volumes, which are unimportant in the steady-state regime. When this option is used, the fluid flow timestep is arbitrarily set equal to the mechanical timestep, and the fluid bulk modulus is automatically defined. The flow time scale has no physical significance, and the same applies to the pressure or flow rate histories before steady state is reached. (3) Transient Flow Analysis Incompressible Fluid (SET flow incompressible) An alternative algorithm has been implemented in UDEC, to make possible transient flow analysis for larger time scales than the basic algorithm allows (see Section 2.2.3). The fluid is assumed to be incompressible, and this scheme uses an iterative procedure within each flow timestep to adjust the joint pressures and domain volumes to ensure flow continuity. To use this option, the user must select SET flow incompressible and define the timestep with SET dtflow. The convergence of the relaxation process in each fluid step is governed by two criteria: (1) the maximum ratio of unbalanced fluid volume to domain volume, set with SET voltol (default = 0.001); and (2) the maximum number of mechanical iterations, defined with SET maxmech (default = 500). To check convergence, PRINT max prints

173 FLUID FLOW IN JOINTS 2-27 out the current maximum unbalanced volume, as well as the actual number of iterations performed in the latest fluid step; HIST unbvol will record this variable. Based on these indicators, the user may adjust the convergence criteria for a specific problem. The purpose of this algorithm is not to simulate dynamic flow or short transients, but the mechanical-fluid coupled response of a system undergoing a quasi-static process. Therefore, at each fluid step, an equilibrium must be possible. Otherwise, the mechanical iteration procedure may not converge. This scheme is not suitable for problems involving extensive failure, or when blocks may become detached. For good performance, sudden loads should not be applied to the system. For example, a cavern should not be pressurized or depressurized instantly, but a finite time should be allowed to elapse. This can be simulated by specifying a history in the PFIX command. For example, domain number 100 can be assigned a pressure of 1 MPa applied over a finite time, as specified by table number 1: pfix press 1e6 hist table 1 range domain 100 where a TABLE is used to define a simple history. The incompressible fluid algorithm is applicable for either confined or unconfined flow conditions. (4) Gas Flow Analysis (SET flow gas) The basic transient compressible flow logic is modified to simulate the special case of highly compressible gas. This logic can be accessed by specifying SET flow gas. Gas density is now dependent upon the pressure in the domain. Gas is considerably more compressible than liquid and, consequently, pressures and stress distributions around gasfilled fractures can be significantly different from those around liquid filled fractures. (5) Two-Phase Flow Analysis (SET flow twophase) The basic transient compressible flow algorithm has also been adapted to simulate twophase fluid flow in fractures. This logic is accessed via the SET flow twophase command. This logic permits the independent specification of pressure and saturation conditions for each fluid phase. The flow calculation can be switched on and off with SET flow on and SET flow off. Note that for all of the compressible fluid modes (i.e., for all options except the incompressible mode), fluid pressures will still change due to joint deformation. Otherwise, they remain constant. Also, flowonly analysis can be performed with all of the compressible fluid modes by selecting SET mech off, in which case joint apertures will be unchanged. incompressible flow mode requires mechanical coupling.

174 2-28 Special Features Structures/Fluid Flow/Thermal/Dynamics Properties Joint Flow Properties Joint conductivity parameters are given with the JOINT command. Their physical meaning is discussed in Section 2.3. The joint permeability factor is assigned by the jperm keyword, the residual aperture by ares, and the nominal aperture at zero normal stress by azero. These properties are required for all five flow calculation modes. In order to modify the permeability relation, the exponent of the cubic law may be changed with expa (default is 3), and the coefficient with empb (default is 1). For two-phase flow in joints, the non-wetting fluid joint permeability is also needed, and is specified with the nwjperm keyword. For jcons = 5 joints, flow through unfractured joints can be turned off with the SET j5flow command. In explicit algorithms, large variations of permeabilities throughout the system generally reduce the numerical efficiency. Joint conductivity is strongly dependent on aperture, due to the cubic exponent. Therefore, for efficiency, a limit on the maximum aperture should be set. The SET capratio command is used to define the maximum aperture ratio allowed (i.e., the maximum hydraulic aperture equals capratio ares). The default value is 5. A large capratio increases the number of cycles necessary to reach steady state (for steady mode) or reduces the fluid timestep (for compressible, gas or twophase mode). Fluid Properties The fluid properties are given with the FLUID command. Fluid density is only required in problems with gravity, to obtain the hydrostatic pressure component. The bulkw keyword is used for the compressible, gas and twophase flow modes. For two-phase flow, the density and bulk modulus of the non-wetting fluid are set with the nwdens and nwbulk keywords, respectively. A threshold fluid gradient to initiate viscoplastic flow can be specified with the cohw keyword. Gas properties are specified in gas flow mode to calculate gas density, ρ g, and bulk modulus, K g. The relation for gas density is and for gas bulk modulus is ρ g = Bp α (2.48) K g = pα (2.49) Note that α is equal to 1/γ (compare Eq. (2.48) to Eq. (2.12) and Eq. (2.49) to Eq. (2.14)). The property keyword gas alpha is used to set α, and the gas constant keyword is used to set B. In addition, minimum values of bulk modulus, gas bulkmin, and density, gas densitymin, are set to avoid numerical problems if the gas pressure drops to zero. Domain Properties A minimum hydraulic aperture is set with the SET capmin command to calculate the minimum domain volume for transient flow analyses. (Note that this does not affect the joint hydraulic conductivity.) If capmin is not specified, it is set to the minimum value for ares. For twophase flow, the capillary pressure curve (Eq. (2.24)) parameters a 0, β and γ are specified with the PROPERTY keywords acap, bcap and gcap, respectively.

175 FLUID FLOW IN JOINTS 2-29 Also, for two-phase flow, a lower bound is specified for saturation of a domain by the wetting fluid, in order to increase the calculation efficiency. This is defined with the SET satmin command. By default, the domain is considered to be fully saturated when the wetting fluid saturation equals one. The value for full saturation can be reduced via the SET satmax command. Temperature Dependency Temperature-dependent fluid density and joint permeability can be specified with the FLUID dtable and PROPERTY ktable commands, respectively, to simulate oneway coupling of thermal and fluid flow processes. The table corresponding to dtable relates water density to temperature. The table corresponding to ktable relates joint permeability factor. The table corresponding to jperm relates to temperature. The TABLE command is used to specify the temperature dependency. If the temperature is not within the range defined by a table, the value for the closest temperature is used (i.e., properties are constant outside the temperature range). Either dtable or ktable, or both, may be specified for an analysis. Default values must be given using FLUID dens or JOINT jperm. The table specified by ktable applies to all joint permeabilities, regardless of joint material number Boundary Conditions Boundary conditions may be applied in terms of fluid pressures (BOUNDARY pp, pxgrad, pygrad), or by defining an impervious boundary (BOUNDARY impermeable). Note that these keywords only supply flow boundary conditions. The mechanical pressure of the fluid in the outer domain must be given independently (with a BOUNDARY stress command, for example). Outer boundary stresses are assumed to be total stresses. Therefore, the example in-situ stress state described in Section is equilibrated by the boundary stresses bound stress ygrad Time-varying fluid pressure can be prescribed for specific boundaries with the hist keyword to the BOUNDARY command. This is done in the same way as discussed for the PFIX command, described in Section Either mechanical or fluid-flow time may be defined as reference time when applying time-varying boundaries with the SET reftime mech or SET reftime flow command, respectively. The default is mech. When applying histories, it is sometimes useful to change the initial time reference. This can be done with SET time (mechanical time) or SET ftime (flow time). This has no effect on the analysis. For two-phase flow, the non-wetting boundary conditions may be applied in terms of non-wetting fluid pressures (BOUNDARY nwpp, nwpxgrad, nwpygrad), or by defining an impervious boundary (BOUNDARY nwimperm). Fluid saturation can be assigned at a boundary (BOUNDARY saturation, satxgrad, satygrad), and a seepage boundary can be set with BOUNDARY seepage. In the latter case, wetting and non-wetting fluid pressures are fixed and equal, and an unsaturated boundary is impermeable for the wetting fluid.

176 2-30 Special Features Structures/Fluid Flow/Thermal/Dynamics A porous medium grid can also be created around the outer boundary of the UDEC model with the FBOUNDARY command. The first mention of FBOUNDARY causes the fluid boundary grid to be created. The fluid boundary must be created for a single convex block (i.e., before any joints are created). The number of fluid boundary zones in a circumferential direction is equal to the number of sides between corners on the single block. (Use the BLOCK angle command to increase the number of corners on the single block.) An example application of FBOUNDARY is given in Section The following keywords define the conditions of the porous medium mesh. Once specified, the conditions cannot be changed later. The number of elements radially around the outer boundary is set by the n keyword (default n = 2). For Figure 2.8, n = 5. The three components of the permeability tensor, K ij, are given by the k11, k12 and k22 keywords. The fluid pressure at the outer boundary of the mesh is specified with pp. Nodes at the outer boundary of the mesh retain this fixed pressure. A pressure gradient can also be specified with pxgrad and pygrad keywords. The pressures at nodes within the mesh are then computed from the equation p = pp + pxgrad x + pygrad y (2.50) where x and y are the coordinates of the node. The radius of the mesh, with origin at the centroid of the block, is set by radius, and the ratio between radial size of adjacent elements is set by rat (default rat = 1). FBOUNDARY is only applicable for steady-state flow. Thus, the SET flow steady command must be given to perform a steady-state analysis. A BOUNDARY command must be given before cycling, because this serves to transfer information between the flow calculations in the rock joints and the flow calculation in the porous medium. For example, the BOUNDARY pp = 0 command will establish the boundary linked list Initialization of Fluid Pressures in Domains The UDEC fluid flow logic is based on the assumption that blocks are impermeable. Block stresses are therefore total stresses, while joint stresses are effective stresses. In order to create a balanced insitu state, the initialization of block/joint stresses and domain pressures must be done consistently. The INSITU command is used for this purpose. The user may define a domain pressure distribution with the pp, pxgrad, pygrad and ywtable keywords. Total stresses are specified with the stress, xgrad and ygrad keywords; these are assigned to the blocks. Joint stresses are calculated by adding the given domain pressures to the block stresses. Note that block compressive stresses are negative, while domain pressures and joint normal compressive stresses are positive. For example, consider the model conditions defined by the commands zone dens fluid dens gravity 0-10

177 FLUID FLOW IN JOINTS 2-31 Assume that the initial water table is located at the free surface at y = 10. A balanced in-situ state is created by the command insitu stress ygrad & pp 0.1 pygrad Alternatively, the same state may be obtained by insitu stress ygrad & ywtable 10 The joint stresses will be calculated automatically to balance the block stresses and domain pressures. The PFIX command also allows the input of domain (wetting and non-wetting) pressures. However, in general, these will not be consistent with the existing block or joint stresses, so the INSITU command is preferred in the establishment of an in-situ state. The PFIX command is normally used to maintain given domain pressures at a fixed value, or to prescribe a pressure history. (The WELL command can also be used to assign fixed flow rates to given domains.) A time-varying pressure may be specified with the PFIX command by using the hist keyword. A seepage condition can be specified on the boundaries of a domain using PFIX seepage. In this latter case, wetting and non-wetting fluid pressures are fixed and equal, and wetting fluid can inflow into the domain only from saturated domains Initialization of Pore Pressures in Blocks UDEC does not model fluid flow through blocks. However, steady-state pore pressures can be assigned to zones within deformable blocks. This is accomplished by adding the zone pp keyword to the end of the INSITU command. By adding the zone pp keyword, poroelastic or poroplastic deformations associated with a new distribution of pore pressures within deformable blocks will be calculated. When this keyword is given with the INSITU command, the pore pressure change multiplied by the Biot coefficient is automatically subtracted from the total normal stresses in the affected zones.* The Biot coefficient, α, relates the compressibility of the grains to that of the drained bulk material: α = 1 K K s (2.51) where K is the drained bulk modulus of the matrix, and K s is the bulk modulus of the grains (see Detournay and Cheng 1993, for reference). The Biot coefficient is assigned with the ZONE model biot coef command. * Pore pressures can also be added to zones by using the INITIAL zone pp command. However, the adjustment to total stress is not performed automatically if the INITIAL command is used to add or change zone pore pressures.

178 2-32 Special Features Structures/Fluid Flow/Thermal/Dynamics For soils, matrix compliance is usually much higher than grain compliance (i.e., 1/K >>> 1/K s ), and it is a valid approximation to assume that the Biot coefficient is equal to 1. For porous rocks, however, matrix and rock compliances are most often of the same order of magnitude and, as a result, the Biot coefficient may be much smaller than 1. Consider, for example, a sample of porous elastic rock. The pores are saturated with fluid at a pressure, p, and a total external pressure, P, is applied around the periphery (i.e., on the outside of an impermeable sleeve). The problem can be analyzed by superposition of two stress states: state a, in which fluid pressure and external pressure are both equal to p; and state b, in which pore pressure is zero, and the external pressure is P p (see Figure 2.10). Figure 2.10 Decomposition of stresses acting on a porous, elastic rock The stress-strain relation for state a may be expressed as For state b (there is no fluid), we can write p = K s ɛ a (2.52) P p = Kɛ b (2.53) The total strain is given by superposition of the strain in state a and in state b: ɛ = ɛ a + ɛ b (2.54) After substitution of ɛ a from Eq. (2.51), and ɛ b from Eq. (2.52), we obtain ɛ = p K s + P p K (2.55)

179 FLUID FLOW IN JOINTS 2-33 After some manipulations, the stress-strain equation takes the form P [1 K K s ] p = P αp = Kɛ (2.56) Clearly then, in the framework of Biot theory, a zero Biot coefficient implies that the elastic stressstrain law becomes independent of pore pressure. Of course, in general, porous rocks do not behave elastically, and pore pressure has an effect on failure. Also, if fluid flow in rocks occurs mainly in fractures, Biot theory may not be applicable. Nonetheless, there are numerous instances where the small value of the Biot coefficient may help explain why pore pressure has little effect on deformation for solid, porous (i.e., unfractured) rocks. (For example, the effect on surface settlement of raising or lowering of the water table in a solid, porous rock mass may be unnoticeable.) Detournay and Cheng (1993) describe laboratory measurements to determine Biot coefficient for saturated rocks. Table 2.3 lists the Biot coefficient along with drained bulk modulus for several rock types. Table 2.3 Poroelastic constants for some rocks [Detournay and Cheng 1993] Rock K (GPa) α Rule sandstone Tennessee marble Charcoal granite Berea sandstone Westerly granite Weber sandstone Ohio sandstone Boise sandstone The logic for grain compressibility, as developed in the framework of Biot theory, is provided in UDEC. A simple verification example is described in Section to illustrate the logic.

180 2-34 Special Features Structures/Fluid Flow/Thermal/Dynamics Solution For flow-only calculations (SET mech off) with compressible flow, the solution time is based upon the fluid timestep and is controlled by the CYCLE n command, where n is the number of fluid flow steps. For a coupled mechanical-flow calculation run to steady state flow (SET flow steady), the SOLVE command can be used to automatically stop the simulation when a steady state is reached. The update of mechanical quantities as a function of fluid pressure occurs by default when pressures change by 1% of the maximum pressure. This can be changed with the SET ptol command. For compressible, gas or twophase fluid-flow mode, the mechanical and fluid timesteps required for stability in a coupled simulation are calculated independently. In a dynamic flow analysis, these timesteps should be the same, so the user should use the FRACTION fb fz fw command to change either the mechanical step (fb or fz) or the fluid step (fw). If a quasi-static approximation is assumed, the mechanical time scale has no significance, so the user may use the SET nfmech command to perform several mechanical steps within each cycle. For gas, steady, compressible or twophase mode, cycle number and time refer to mechanical steps and mechanical time. For steady mode, the fluid timestep is set equal to the mechanical timestep, since the fluid time scale is arbitrary. Convergence to steady state can be checked with the total inflow/outflow indicators given by PRINT max. When using incompressible mode, the timestep must be defined by the user with SET dtflow. In the CYCLE command, either the number of fluid steps or the flow time increment is given (with CYCLE ftime). The mechanical cycle number and the total flow time are printed to the screen when the CYCLE command is issued. For coupled, thermal simulations, if SET flow steady is specified, the fluid-flow timestep has no physical meaning; thermal time is the only relevant time measure. However, if a transient incompressible flow calculation is performed (SET flow incompressible), then the thermal and hydraulic times should be consistent. Use the SET thdt dt t command to define the thermal timestep, and SET dtflow dt f to define the fluid-flow timestep. The number of thermal steps is specified by SET nther nt, and the number of fluid flow steps by SET nflow nf. When the RUN step = ns command is invoked, the program alternates between nt thermal timesteps and nf flow timesteps until the number of thermal timesteps reaches ns. In order for fluid-flow and thermal times to be consistent, the following condition must be satisfied. nt x dt t = nf x dt f If this condition is not fulfilled, UDEC issues a warning and automatically modifies dt t to satisfy the condition.

181 FLUID FLOW IN JOINTS Output Options The PRINT command keywords domain, contact, flow and max list the main flow-related variables. Graphical output is obtained with the PLOT command keywords pp, flow, vflow and aperture. Pressures and flow rates along a joint may be printed or plotted with the PRINT joint or PLOT joint command, or across joints with the PLOT jline command. Histories of flow rates at a particular contact may be recorded with HISTORY flowrate n, where n is the contact address. Domain pressure histories are requested with HISTORY pp xy; the domain closest to (x,y) is selected. Note that when using the steady option, the flow time scale has no physical meaning, and these histories are only worth checking if steady-state conditions have been reached. For the compressible or incompressible flow mode, the flow time should be recorded in a history (with HISTORY flowtime) so that the other histories may be plotted against it, since mechanical time has no real significance.

182 2-36 Special Features Structures/Fluid Flow/Thermal/Dynamics 2.5 Verification and Example Problems Several verification problems and example exercises are presented to illustrate fluid-flow modeling in UDEC. The data files for these examples are located in the Datafiles\Fluid directory Heave of a Rock Layer The logic for grain compressibility in a saturated rock, as developed in the framework of Biot theory, is provided in UDEC. To illustrate this, we consider a layer of sandstone of large lateral extent and thickness H = 500 meters, resting on a rigid base. The layer is elastic, the drained bulk modulus of the rock, K, is 13 GPa, and the shear modulus, G, is 8.43 GPa. The bulk density of the dry rock is 2000 kg/m 3, and the density of water, ρ w, is 1000 kg/m 3. The porosity of the rock, n, is uniform with a value of 0.1. Gravity, g, is set to 10 m/sec 2. The rock is initially dry. The water table is then raised to the rock surface, which induces heave in the rock. The surface heave can be evaluated analytically using Eq. (2.57), where α 1 = K + 4G/3. u h = (n α)ρ wg 2α 1 H 2 (2.57) A single columnar shaped block is used for this exercise. The block is 500 meters in the y-direction and 10 meters in the x-direction. The origin of axes is at the bottom of the model. The mechanical boundary conditions correspond to roller boundaries at the base and lateral sides of the model. We first consider equilibrium of the dry layer. We initialize the stresses using the INSITU stress command (without the zone pp keyword), and use a value of (= (K 2G/3)/(K + 4G/3)) for the coefficient of earth pressure at rest, k o. There are two competing effects on deformation associated with raising the water level: first, the increase in pore pressure will generate heave of the layer; second, the increase in rock bulk density due to the presence of water in the pores will induce settlement. We proceed to model the combined effects on deformation of a rise in water level up to the rock surface as follows. We assume the sandstone deformational behavior is similar to Weber sandstone (see Table 2.3) and set the Biot coefficient to 0.64 using the ZONE model biot coef command. We specify a hydrostatic pore-pressure distribution corresponding to a new water level at the top of the model by using either the INSITU ywtable command or the INSITU pp command with the pygrad keyword to specify the pore-pressure gradient. The zone pp keyword is also given with both commands. A total stress correction will now be applied automatically when the pore pressure is changed with the INSITU command. It is also necessary to specify a wet (or saturated) density for the rock below the water table. The saturated density, ρ s, is calculated from the equation

183 FLUID FLOW IN JOINTS 2-37 ρ s = ρ d + nρ w (2.58) where ρ d is the bulk density of the dry rock, n is the porosity of the rock and ρ w is the density of water. For the given properties in this example, the saturated density of the rock is 2100 kg/m 3. This density is specified with the ZONE model density command. The data file for this simulation is listed in Example 2.1. Using Eq. (2.57), the heave of the rock after raising the water table is calculated to be meters. This compares well with the displacement calculated by UDEC. Figure 2.11 plots displacement vectors that show the heave for this model. Note that if the Biot effect is not included (i.e., α assumed equal to 1), then the calculated heave is meters. JOB TITLE : (*10^2) UDEC (Version 5.00) LEGEND Feb :40:38 cycle 3471 time 1.575E+00 sec flow time = 1.575E+00 sec block plot displacement vectors maximum = 2.810E E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^2) Figure 2.11 Heave of a rock layer

184 2-38 Special Features Structures/Fluid Flow/Thermal/Dynamics Example 2.1 Heave of a rock layer new ;file heave.dat config fluid set flow clear steady off round 0.1 edge 0.2 block 0,0 0,500 10,500 10,0 gen quad 10.0 group zone sandstone zone model elastic dens 2000 bulk 13e9 shear 8.43e9 range group sandstone fluid density= boundary xvelocity 0 range -0.1, ,500.1 boundary xvelocity 0 range 9.9, ,500.1 boundary yvelocity 0 range -0.1, ,0.1 set gravity=0-10 history ydisplace 5.0,250.0 history ydisplace 5.0,500.0 insitu stress ,0.0, xgrad 0.0,0.0,0.0 ygrad & ,0.0, szz zgrad 0.0, history unbalanced solve ratio 1.0E-5 save h1.sav zone density 2.1E3 range group sandstone zone biot 0.64 range group sandstone insitu ywtable 500 zone_pp ; or install pore pressure distribution with INSITU PP ; insitu pp pygrad zone_pp ; reset disp solve ratio 1.0E-5 save h2.sav

185 FLUID FLOW IN JOINTS Example Application of the Fluid Boundary In order to validate the fluid boundary logic, the mesh shown in Figure 2.8 was applied as a fluid boundary for the UDEC block model shown in Figure JOB TITLE : Fluid Boundary Test UDEC (Version 5.00) (*10^1) LEGEND 8-Sep :18:32 cycle 500 time 6.769E-02 sec block plot Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 2.12 UDEC model for fluid boundary example The joint characteristics for this example are aperture joint permeability factor spacing constant and equal to 10 4 m 100 Pa 1 s 1 1 m

186 2-40 Special Features Structures/Fluid Flow/Thermal/Dynamics Two joint sets, with orientations 20 and 80 from the x-axis, are generated. The equivalent permeability tensor for any continuous joint set with spacing, s, orientation, α, from the x-axis, aperture, a, and joint permeability factor, k j, can be derived: where C is the conductivity of a joint. C = k j a 3 (2.59) K j = C s = k j a 3 s (2.60) where K j is the equivalent permeability in the direction of the joint set. K 11 = K j cos 2 α K 22 = K j sin 2 α (2.61) K 12 = K j sin α cos α where K 11, K 22 and K 12 are the three components of the equivalent permeability tensor. If several joint sets are superposed, their contributions to the permeability tensor should be summed. In our example, this yields K 11 = m 3 /(Pa s) K 22 = m 3 /(Pa s) K 12 = m 3 /(Pa s) A horizontal pressure gradient, p = x, in Pa/m is imposed on the outer fluid boundary. The corresponding data file is listed in Example 2.2. This problem is run as an uncoupled analysis. First, a flow-only run is made to steady state, and then a mechanical-only run is made. The fluid pressure field in the model at steady state is shown in Figure 2.13.

187 FLUID FLOW IN JOINTS 2-41 JOB TITLE : Fluid Boundary Test (*10^1) UDEC (Version 5.00) LEGEND 8-Sep :42:35 cycle 4041 time 5.471E-01 sec domain pore pressures maximum pressure= 4.500E+05 each line thick = 9.000E E E E E E E E E E E E+05 Fluid Boundary Grid Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 2.13 Fluid pressure field Example 2.2 Application of the fluid boundary logic ; test of fluid boundary logic config fluid set flow clear steady on round 0.01 edge 0.02 block (ang=15) 0,0 0,10 10,10 10,0 fboundary k E-11 k E-11 k E-10 pp pxgrad & rad 20.0 rat 1.2 n 5 jset angle 20 spacing 1 origin 0,0 jset angle 80 spacing 1 origin 0,0 delete area 0.01 jdelete gen edge 1.0 group zone rock zone model elastic density 1E3 bulk 1E9 shear 5E8 range group rock group joint joint joint model area jks 1E8 jkn 1E8 jfriction 45 jcohesion 1E4 jtension 1E4 & jperm 100 ares 1E-4 azero 1E-4 range group joint set jmatdf=1 prop jmat=1 jks=1e8 jkn=1e8 jfriction=45 jcohesion=1e4 jtension=1e4 & jperm=100 ares=1e-4 azero=1e-4 fluid density=1000.0

188 2-42 Special Features Structures/Fluid Flow/Thermal/Dynamics boundary stress ,0.0, boundary pp pygrad ; mode=-1 insitu stress ,0.0, szz pp 20e4 pxgrad 0 & pygrad -1e4 zone_pp set gravity= history pp 0.0,0.0 history pp 10.0,10.0 set capratio=1.0 set mech=off history unbalanced ; cycle with fluid boundary to steady state (no mechanical) cycle 500 save sf1.sav boundary xvelocity 0 range -0.1, ,10.1 boundary xvelocity 0 range 9.9, ,10.1 boundary yvelocity 0 range -0.1, ,0.1 set mech=on set flow=off ; cycle to mechanical equilibrium (no flow) solve ratio 1.0E-5 save sf2.sav set mech=off set flow=on ; check that fluid is at steady state cycle 1000 save sf3.sav ;*** plot commands **** ;plot name: block plot hold block ;plot name: fluid pressure plot hold pp fboundary ; plot name: pp histories plot hold history 1 2

189 FLUID FLOW IN JOINTS Steady-State Fluid Flow with Free Surface The fluid flow logic in UDEC allows for situations with a free surface in addition to confined flow problems. This test compares the UDEC results with a simple analytical solution for 2D flow in a homogeneous aquifer governed by Darcy s Law: where v = discharge velocity; k = coefficient of permeability (length / time); and i = hydraulic gradient. The problem is shown in Figure 2.14: v = ki (2.62) h 1 h 2 L impermeable base Figure 2.14 Figure showing definition of terms in Dupuit s formula Dupuit s formula (see, for example, Harr 1962, p. 42) gives the total discharge (per unit width) as Q = k h2 1 h2 2 2 L (2.63) The UDEC model is shown in Figure 2.15 (block plot). A system of two sets of joints with constant aperture (i.e., stress-independent) and constant joint spacing, s, was used to simulate the homogeneous isotropic medium.

190 2-44 Special Features Structures/Fluid Flow/Thermal/Dynamics The dimensions are L =8m h 1 =4m h 2 =1m s = 0.5 m JOB TITLE : Flow with a Free Surface UDEC (Version 5.00) LEGEND 4-Oct :39:42 cycle 1000 time 2.041E+02 sec block plot Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.15 UDEC problem geometry for verification of fluid flow logic The flow rate in a single joint of length, L, subject to a pressure difference of P, is calculated in UDEC as q = k j a 3 P L (2.64) where k j = (1/12μ); μ = dynamic viscosity; and a = aperture.

191 FLUID FLOW IN JOINTS 2-45 For a system of joints with spacing, s, the average velocity for an equivalent porous medium would be v = q s = 1 P a3 12 μ L = 1 12 μ a3 ρ w g h L 1 S 1 S (2.65) where ρ w g is the fluid mass density (e.g., ρ w = 1000 kg/m 3 for water); and is the gravitational acceleration. Because h/l is the hydraulic gradient i in Darcy s Law, the coefficient k corresponds to k = ρ w g 12 μ a 3 S (2.66) For the conditions a = m ρ w = 1000 kg/m 3 g = 10 m/sec 2 S = 0.5 m k j = 12 1 μ = 83.3 Pa 1 sec 1 we have k = m/sec. Then, Dupuit s formula gives Q = m 3 /sec The UDEC model gives (from the sum of discharge flow rates given by the PRINT max command) The error is 6.4%. Q = m 3 /sec

192 2-46 Special Features Structures/Fluid Flow/Thermal/Dynamics Figure 2.16 shows the flow rates for the steady state. Figure 2.17 shows the domain pressures. JOB TITLE : Flow with a Free Surface UDEC (Version 5.00) LEGEND 4-Oct :39:42 cycle 1000 time 2.041E+02 sec boundary plot flow rates max flow rate = 6.949E-07 each line thick = 1.390E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.16 UDEC steady-state flow rates JOB TITLE : Flow with a Free Surface UDEC (Version 5.00) LEGEND 4-Oct :39:42 cycle 1000 time 2.041E+02 sec boundary plot domain pore pressures maximum pressure= 3.623E+04 each line thick = 7.246E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.17 UDEC domain pressures

193 FLUID FLOW IN JOINTS 2-47 Example 2.3 Steady-state fluid flow with a free surface config fluid set flow clear steady on round 0.01 edge 0.02 block -4,0-4,4 4,4 4,0 jset angle 0 spacing 0.5 origin 0,0 jset angle 90 spacing 0.5 origin 0,0 change mat 1 property mat 1 density 1 group joint joint joint model area jks 1 jkn 1 jperm 83.3 ares azero range & group joint ; new contact default set jcondf joint model area jks=1 jkn=1 jperm=83.3 ares=1e-4 azero=1e-4 fluid density= set gravity=0-10 boundary pp pygrad range -4.01, ,4.1 boundary pp pygrad range 3.99, ,1.1 boundary impermeable range -4.1, ,0.1 history pp -4.0,2.0 history pp 4.0,0.0 set mech=off cycle 1000 save steady.sav

194 2-48 Special Features Structures/Fluid Flow/Thermal/Dynamics Pressure Distribution in a Fracture with Uniform Permeability (Aperture) In this problem, time variations of pressure distribution in a fracture are examined as a result of a fluid pressure, P o, being suddenly applied to one end of a fracture (see Figure 2.18). Crack with Uniform Constant Aperture P 0 Impermeable l X Figure 2.18 Model geometry and boundary conditions The pressure in a finite length fracture that is suddenly pressurized at one end can be found from an analogous solution given for a 1D heat conduction problem. (See, for example, Hardy and Asgian 1989.) The solution satisfies the differential equation ( α 2 12 μ )( d 2 P dx 2 ) = β dp dt (2.67) where α is the hydraulic aperture (m); β is the fluid compressibility (Pa 1 ); and μ is the dynamic viscosity (Pa/sec). The values selected for this problem are l =1m P o = 9.5 MPa α = m β = Pa 1 μ =10 3 Pa sec A dimensionless time, T, for Eq. (2.67) can be defined as

195 FLUID FLOW IN JOINTS 2-49 T = (α2 / 12 μ) t βl 2 (2.68) The solution of Eq. (2.67) with the boundary conditions defined in Figure 2.18 is given by Carslaw and Jaeger (1959): P P o = π [ n=0 e (2n+1)2 (T /4)π 2 ( (2n + 1) π cos 2 ζ ) ( ( 1) n+1 2n + 1 )] (2.69) where T = the dimensionless time; ζ = x/l (see Figure 2.18); P = pore pressure at a distance x from the impermeable side; and P o = pore pressure at a distance x = l. The fluid flow properties are set in UDEC: bulkw =1/β =20MPa jperm =1/(12 μ) = MPa 1 s 1 The hydraulic aperture is maintained at a constant value of α by setting the UDEC parameters azero ares = α = m = α = m Example 2.4 contains the data file for this problem. Figure 2.19 compares both the analytical solution and the UDEC solution for three values of dimensionless time, T. The FISH function, ana sol, calculates the analytical solution for times, T = 0.1, 0.3 and 0.5, and stores the results in Tables 2, 4 and 6, respectively. The function, num sol, collects the UDEC results in Tables 1, 3 and 5 for comparison to the analytical results.

196 2-50 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE :. UDEC (Version 5.00) 1.00 LEGEND 4-Oct :00:00 cycle 518 time 3.333E-01 sec flow time = 3.333E-01 sec table plot Time = UDEC X X X Time = 01 - Analytic Time = UDEC X X X Time = 03 - Analytic Time = UDEC X X X Time = 05 - Analytic Vs. 6.30E-02<X value> 9.37E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.19 Comparison of analytical and UDEC solutions for joint fluid pressure (P / P o ) at various distances (x/l) in a fracture with zero initial pressure at x =0 Example 2.4 Pressure distribution in a fracture with uniform permeability def constants tabo = -1 tabe = 0 length = 1.0 p_0 = 9.5 alpha = 3.0e-5 beta = 5.0e-8 dy_vis = 1.0e-3 t_cons = (alpha * alpha) / (12.0 * dy_vis * beta * length * length) overl = 1.0 / length n_max = 100 fop = 4.0 / pi teps = 1.0e-4 end constants def num_sol tabo = tabo + 2 pnt = domain_head loop while pnt # 0 x_pos = length - d_x(pnt)

197 FLUID FLOW IN JOINTS 2-51 if d_vol(pnt) > 0.0 then table(tabo,x_pos) = d_pp(pnt) / p_0 endif pnt = d_next(pnt) endloop end def ana_sol tabe = tabe + 2 t_cap = t_cons * ftime pnt = domain_head loop while pnt # 0 if d_vol(pnt) > 0.0 then x_pos = d_x(pnt) x_l = x_pos * overl n = 0 tsum = 0.0 tsumo = 0.0 sgnn = -1.0 converge = 0 loop while n < n_max fn = float(n) n = n + 1 tnp1 = 2.0 * fn term1 = exp(-0.25*(tnp1ˆ2)*t_cap*pi*pi) term2 = cos(0.5*tnp1*pi*x_l) term3 = sgnn / tnp1 term = term1 * term2 * term3 tsum = tsumo + term if abs(tsum - tsumo) < teps then table(tabe,x_pos) = fop * tsum converge = 1 n = n_max else tsumo = tsum endif sgnn = -1.0 * sgnn endloop if converge = 0 then ii = out( not converged x= + string(x_pos) + t= + string(ftime)) exit endif endif pnt = d_next(pnt) endloop end ;

198 2-52 Special Features Structures/Fluid Flow/Thermal/Dynamics config fluid set flow clear compressible on round edge block 0,0 0,0.1 1,0.1 1,0 crack (0,0.05) (1,0.05) gen edge 0.2 group zone dummy zone model elastic density 1 bulk 1 shear 1 range group dummy group joint joint joint mod area jks 1 jkn 1 jperm 8.33E7 ares 3E-5 azero 3E-5 range group & joint ; new contact default set jcondf joint model area jks=1 jkn=1 jperm=8.33e7 ares=3e-5 azero=3e-5 fluid density= fluid bulkw=20.0 boundary pp 9.5 range , ,0.101 boundary impermeable range 0.999, ,0.101 ; ; print status to find out fluid flow time step, t, ; for the determination of dimensionless time T ; cycle 0 print info set mech off ; ; flow time = (T = 0.1) cycle 104 num_sol ana_sol save time1a.sav ; ; flow time = 0.20 (T = 0.3) cycle 207 num_sol ana_sol save time2a.sav ; ; flow time = 0.33 (T = 0.5) cycle 207 num_sol ana_sol save time3a.sav label tabl 1 Time = UDEC label tabl 2

199 FLUID FLOW IN JOINTS 2-53 Time = Analytic label tabl 3 Time = UDEC label tabl 4 Time = Analytic label tabl 5 Time = UDEC label tabl 6 Time = Analytic pl table 1 cross 2 3 cross 4 5 cross 6 ret

200 2-54 Special Features Structures/Fluid Flow/Thermal/Dynamics Transient Fluid Flow in a Single Joint in an Elastic Medium A simple example of a single joint embedded in an elastic medium was used to test the incompressible flow algorithm (SET flow incompressible) and the compressible flow algorithm (SET flow compressible). Figure 2.20 shows the problem geometry. The joint extends from x = 0 to x = 10 m, and is discretized into 10 segments. The 2-m end segments on either side are assumed to be fictitious (construction) joints (i.e., high strength and impermeable). Flow is injected at a constant rate of m 3 /sec into the left of the actual joint (A). Pressure is held at the in-situ value p o at the right end (D). The following properties were used. Blocks Young s modulus E = Pa Poisson s ratio v = 0.25 Joint normal and shear stiffness permeability factor initial aperture (at zero stress) residual aperture k n = k s = Pa/m k j = 300 Pa 1 sec 1 a o = m a res = m In-situ stresses vertical stress fluid pressure σ yy = Pa p o = Pa Simulations with fluid timesteps of 1 second and 10 seconds were performed. Figure 2.21 shows the evolution of the pressure at the domains A, B and C during the first 100 seconds, obtained in the run with the smaller timestep. Figure 2.22 shows the corresponding curves for a timestep of 10 seconds. It can be seen that the results agree very well, except for the initial transient, which obviously cannot be represented with the larger timestep. This second run was continued until steady state was reached, as shown in Figure As a comparison, the same problem was run with the SET flow compressible option. A fluid bulk modulus, K w, of 20 MPa was assumed. This is about 1/100 of the real modulus for pure water (i.e., no entrained air). But, given the joint apertures, the effective fluid stiffness (K w /a) is on the order of (i.e., two orders of magnitude higher than the joint normal stiffness). The pressure histories at the same three points, displayed in Figures 2.24 and 2.25, are very close to those from the run with SET flow incompressible (Figures 2.21 and 2.23), indicating that the assumption of fluid incompressibility is justified for this type of problem.

201 FLUID FLOW IN JOINTS 2-55 The incompressible simulation with the 1 second timestep was run for 100 fluid steps. The total number of mechanical steps performed was The compressible simulation, however, required a much smaller fluid timestep for numerical stability. Initially, the calculated timestep was sec.; subsequently, as the joint opened, the timestep was gradually reduced to a value of sec. Thus, the 100-second simulation required about 10 4 fluid steps. To agree with the assumption of a quasi-static process implied in the incompressible option, 10 mechanical iterations were performed per cycle (i.e., SET nfmech = 10 was used). Therefore, a total of about 10 5 mechanical steps was required in the compressible run, showing that, in practice, the consideration of fluid compressibility is only possible for short-term simulations. Example 2.5 contains the data file for the incompressible flow calculation. Example 2.6 contains the data file for the compressible flow calculation. Figure 2.20 UDEC model for incompressible flow in a single joint

202 2-56 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE :. UDEC (Version 5.00) (e+005) 4.50 LEGEND 4-Oct :01:00 cycle 1631 time 1.818E-01 sec flow time = 1.020E+02 sec history plot Fluid Pressure at A Fluid Pressure at B Fluid Pressure atc Vs. fluid flow time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+002) Figure 2.21 Fluid pressure histories at A, B and C for a 1-second fluid timestep JOB TITLE :. UDEC (Version 5.00) (e+005) 4.50 LEGEND 4-Oct :02:00 cycle 665 time 7.414E-02 sec flow time = 1.200E+02 sec history plot *Fluid Pressure at A *Fluid Pressure at B *Fluid Pressure atc Vs. *fluid flow time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+002) Figure 2.22 Fluid pressure histories at A, B and C for a 10-second fluid timestep

203 FLUID FLOW IN JOINTS 2-57 JOB TITLE :. UDEC (Version 5.00) (e+005) 6.00 LEGEND 4-Oct :03:00 cycle 1994 time 2.223E-01 sec flow time = 1.040E+03 sec history plot **Fluid Pressure at A **Fluid Pressure at B **Fluid Pressure atc Vs. **fluid flow time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+003) Figure 2.23 Fluid pressure histories at A, B and C for a 10-second fluid timestep; run to steady-state flow JOB TITLE :. UDEC (Version 5.00) (e+005) 4.50 LEGEND 4-Oct :05:00 cycle time 8.695E+00 sec flow time = 1.007E+02 sec history plot *Fluid Pressure at A *Fluid Pressure at B *Fluid Pressure at C Vs. **fluid flow time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+002) Figure 2.24 Fluid pressure histories at A, B and C for compressible flow

204 2-58 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE :. UDEC (Version 5.00) (e+005) 5.50 LEGEND 4-Oct :06:00 cycle time 1.826E+02 sec flow time = 1.047E+03 sec history plot Fluid Pressure at A Fluid Pressure at B Fluid Pressure at C Vs. *fluid flow time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+003) Figure 2.25 Fluid pressure histories at A, B and C for compressible flow; run to steady-state Example 2.5 Incompressible transient flow in a single joint in an elastic medium ; ; run with SET FLOW INCOMPRESSIBLE option ; config fluid set flow incompressible on round edge block -2,-5-2,5 12,5 12,-5 crack (-2,0) (12,0) gen quad 1.01,2.51 group zone block zone model elastic density 1E3 bulk E9 shear 2E9 range group block join_cont range -2.1, ,0.1 join_cont range 9.9, ,0.1 ; crack group joint crack joint model area jks 2E10 jkn 2E10 jcohesion 1E20 jtension 1E20 jperm 300 & ares azero 1.7E-4 range group crack ; set new contact default set jmatdf=1

205 FLUID FLOW IN JOINTS 2-59 prop jmat=1 jks=2e10 jkn=2e10 jcohesion=1e20 jtension=1e20 jperm=300 & ares= azero=1.7e-4 fluid density= insitu stress 0.0,0.0, pp 1e5 boundary xvelocity 0 range -2.1, ,-4.9 boundary yvelocity 0 range -2.1, ,-4.9 boundary yvelocity 0 range -2.1, ,5.1 boundary xvelocity 0 range -2.1, ,5.1 ; sink at x = 9.5 pfix ppressure 1E5 range 9.4, ,0.1 ; well at x = 0.5 pfix ppressure 1E5 range -2.1, ,0.1 ; keep insitu pp in impermeable joints pfix ppressure 1E5 range 9.9, ,0.1 well flow 5.0E-7 atdomain (0.5,0.0) history flowtime history pp 0.5,0.0 history pp 2.5,0.0 history pp 4.5,0.0 set capratio=100.0 set dscan= ; turn off scan for new contacts to speed calculation history unbvol save ft0.sav ; 1 second timestep set maxmech=100 set dtflow=1.0 cycle ftime 100 save ft1.sav ; 10 second timestep restore ft0.sav history ncyc 1 set maxmech=100 set dtflow=10.0 cycle ftime 100 save ft10.sav ; run to steady state cycle ftime 900 save ft100.sav

206 2-60 Special Features Structures/Fluid Flow/Thermal/Dynamics Example 2.6 Compressible transient flow in a single joint in an elastic medium ; ; run with SET FLOW COMPRESSIBLE option ; config fluid set flow compressible on round edge block -2,-5-2,5 12,5 12,-5 crack (-2,0) (12,0) gen quad 1.01,2.51 group zone block zone model elastic density 1E3 bulk E9 shear 2E9 range group block join_cont range -2.1, ,0.1 join_cont range 9.9, ,0.1 ; crack group joint crack joint model area jks 2E10 jkn 2E10 jcohesion 1E20 jtension 1E20 jperm 300 & ares azero 1.7E-4 range group crack ; set new contact default set jmatdf=1 prop jmat=1 jks=2e10 jkn=2e10 jcohesion=1e20 jtension=1e20 jperm=300 & ares= azero=1.7e-4 fluid density= fluid bulkw=2.0e7 insitu stress 0.0,0.0, pp 1e5 boundary xvelocity 0 range -2.1, ,-4.9 boundary yvelocity 0 range -2.1, ,-4.9 boundary yvelocity 0 range -2.1, ,5.1 boundary xvelocity 0 range -2.1, ,5.1 ; sink at x = 9.5 pfix ppressure 1E5 range 9.4, ,0.1 ; well at x = 0.5 well flow 5.0E-7 atdomain (0.5,0.0) ; keep insitu pp in impermeable joints pfix ppressure 1E5 range -2.1, ,0.1 pfix ppressure 1E5 range 9.9, ,0.1 history flowtime history pp 0.5,0.0 history pp 2.5,0.0 history pp 4.5,0.0 history ncyc 100 set capratio=100.0 set dscan= ; turn off scan for new contacts to speed calculation

207 FLUID FLOW IN JOINTS 2-61 save fc0.sav ; set nfmech=10 cycle save fcc100.sav ; cycle save fcss.sav

208 2-62 Special Features Structures/Fluid Flow/Thermal/Dynamics Transient One-Dimensional Gas Flow Chan et al. (1993) present the solution for a transient one-dimensional gas flow problem: the timeevolution of the pressure of an ideal gas draining from a joint into a cavity. The problem conditions involve one-dimensional flow along the joint at constant volume in a semi-infinite domain. At x = 0, the gas pressure p = 0, and at x =, the gas pressure is set to p 1. The authors present both an approximate analytical solution and an empirical fit to a numerical solution obtained with an implicit time-marching scheme. They derive an approximate analytical solution for gas pressure along the joint: They also present an empirical fit to a numerical solution: 1 exp( ξ/ 2 ξ 2 /4) (2.70) = 1 e (0.625ξ+0.186ξ 2 ) (2.71) In Eqs. (2.70) and (2.71), dimensionless parameters = p 2 /p 2 1, and ξ = (νnx 2 )/(tkp 1 ), where ν is the shear viscosity of the gas, n is porosity, k is permeability and t is time. The solutions are converted to a FISH function with results written to tables for comparison to UDEC results. This FISH function, ppan, is listed in Example 2.7: Example 2.7 Analytical solution and empirical fit for 1D gas flow def ppan ; analytical solution at time=0.5 ; table 8 : pp vs. x ; table 28 : psi vs. ksi ; table 10 : empirical pp vs. x ; table 30 : empirical psi vs. ksi ; ; analytical solution at time=2.0 ; table 9 : pp vs. x ; table 29 : psi vs. ksi ; table 11 : empirical pp vs. x ; table 31 : empirical psi vs. ksi ; ; fac = k/(visc.*poros.) = (jperm*a3) / poros ; poros = a ; fac = 1e9 * 1e-12 / 1e-4 = 10 ; fac = 10. pp1 = 1.0 tt = 0.5

209 FLUID FLOW IN JOINTS 2-63 loop j (1, 10) xx = (j-1)*1.0 ksi = (1.0/fac) * xx*xx / (tt * pp1) ksi = sqrt(ksi) psi = exp(-ksi/sqrt(2.0)-0.25*ksi*ksi) ppj = sqrt(psi*pp1*pp1) xtable(28,j) = ksi ytable(28,j) = psi xtable(8,j) = xx ytable(8,j) = ppj psi2 = exp(-0.625*ksi-0.186*ksi*ksi) ppj2 = sqrt(psi2*pp1*pp1) xtable(30,j) = ksi ytable(30,j) = psi2 xtable(10,j) = xx ytable(10,j) = ppj2 endloop ; tt = 2.0 loop j (1, 10) xx = (j-1)*1.0 ksi = (1.0/fac) * xx*xx / (tt * pp1) ksi = sqrt(ksi) psi = exp(-ksi/sqrt(2.0)-ksi*ksi/4.0) ppj = sqrt(psi*pp1*pp1) xtable(29,j) = ksi ytable(29,j) = psi xtable(9,j) = xx ytable(9,j) = ppj psi2 = exp(-0.625*ksi-0.186*ksi*ksi) ppj2 = sqrt(psi2*pp1*pp1) xtable(31,j) = ksi ytable(31,j) = psi2 xtable(11,j) = xx ytable(11,j) = ppj2 endloop end The UDEC model contains a single horizontal joint, 10 m long, divided into 10 domains. The model geometry is shown in Figure 2.26.

210 2-64 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE :. UDEC (Version 5.00) LEGEND Oct :20:00 cycle 0 time 0.000E+00 sec flow time = 0.000E+00 sec block plot contact normals corner-corner corner-edge edge-corner Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 2.26 UDEC model for 1D gas flow At x = 0, gas pressure is fixed at zero. At x = 10, the pressure is set to the value of p defined by Eq. (2.71) for p 1 = 1 MPa. The boundary condition is controlled by applying pressure with a FISH history multiplier, ppanx10 (see Example 2.8). The problem parameters are k =10 9, ν =10 12 and n =10 4. Example 2.8 Analytical solution for pressure at x =10 def ppanx10 ; analytical solution at x=10 ; fac = k/(visc.*poros.) = (jperm*a3) / poros ; poros = a ; fac = 1e9 * 1e-12 / 1e-4 = 10 fac = 10. pp1 = 1.0 tt = ftime xx = 10.0 ppj = 1.0 if tt > 0.0 ksi = (1.0/fac) * xx*xx / (tt * pp1) ksi = sqrt(ksi) psi2 = exp(-0.625*ksi-0.186*ksi*ksi) ppj2 = sqrt(psi2*pp1*pp1)

211 FLUID FLOW IN JOINTS 2-65 psi = psi2 ppj = ppj2 endif ppanx10 = ppj end The UDEC data file is listed in Example 2.9. A comparison of the UDEC results to the analytical solution (Eq. (2.70)) and the empirical fit (Eq. (2.71)) at time = 0.5 is shown in Figure 2.27, and at time = 2.0 in Figure Example 2.9 Transient one-dimensional gas flow config fluid set flow clear compressible gas on round 1E-3 edge 2E-3 block 0,-1 0,1 10,1 10,-1 crack (0,0) (10,0) gen quad 1.01,1.01 range 0,10 0,1 gen quad 10.01,1.01 range 0,10-1,0 group zone block zone model elastic density 2.7E-3 bulk shear range group & block group joint joint joint model area jks 1 jkn 1 jcohesion 1E20 jtension 1E20 jperm 1E9 ares & 5E-5 azero 1E-4 range group joint ; new contact default set jcondf joint model area jks=1 jkn=1 jcohesion=1e20 jtension=1e20 & jperm=1e9 ares=5e-5 azero=1e-4 fluid gas_bulk= fluid gas_const=1.0 boundary xvelocity 0 boundary yvelocity 0 boundary pp 0.0 range -0.1, ,0.1 boundary pp 1.0 range 9.9, ,0.1 insitu pp 1 fraction history flowtime history pp 0.5,0.0 history pp 1.5,0.0 history pp 4.5,0.0 history pp 8.5,0.0 history pp 9.5,0.0 history flowrate 0.0,0.0

212 2-66 Special Features Structures/Fluid Flow/Thermal/Dynamics history flowrate 5.0,0.0 history flowrate 10.0,0.0 call ppan.fis ppan call ppanx10.fis ppanx10 boundary pp 1.0 history=ppanx10 range 9.9, ,0.1 call pp1dini.fis pp1dini call getpp1d.fis getpp1d set mech=off cycle 80 call getpp1d.fis getpp1d save gg12ga.sav ; cycle 240 call getpp1d.fis set jjtab=4 getpp1d save gg12g2a.sav ; ;*** plots **** ;plot name: gas pressure time=0.5 plot hold table 23 cross 28 line 30 line ;plot name: gas pressure time= 2.0 plot hold table 24 cross 29 line 31 line

213 FLUID FLOW IN JOINTS 2-67 JOB TITLE :. UDEC (Version 5.00) 1.00 LEGEND 4-Oct :25:00 cycle 80 time 4.991E-01 sec flow time = 4.991E-01 sec table plot UDEC X X X Eq (approx. anal.) Eq (num. fit) Vs. 2.24E-01<X value> 4.25E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.27 Gas pressure along joint ( versus ξ) at time = 0.5 JOB TITLE :. UDEC (Version 5.00) 1.00 LEGEND 4-Oct :30:00 cycle 320 time 1.998E+00 sec flow time = 1.998E+00 sec table plot UDEC X X X Eq (approx. anal.) Eq (num. fit) Vs. 1.12E-01<X value> 2.13E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.28 Gas pressure along joint ( versus ξ) at time = 2.0

214 2-68 Special Features Structures/Fluid Flow/Thermal/Dynamics Filling of a Horizontal Joint No Capillary Effects In this two-phase flow example, the water is injected at one side (at constant pressure P 0 )ofa horizontal joint filled with air. The air is at atmospheric pressure. The analytical solution for the location of the interface X(t) at time t is (Voller et al. 1996) X(t) = (2Pt) where P = kp 0 ; k is the permeability of the joint. JOB TITLE : Filling Joint with Water UDEC (Version 5.00) LEGEND Oct :06:21 cycle 0 time 0.000E+00 sec flow time = 0.000E+00 sec block plot Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.29 Geometry of the UDEC model The problem is simulated using the UDEC model shown in Figure The model is2mby2m in size, and has a single, horizontal joint. It was assumed that capillary forces are zero. The initial saturation of joint is 1%, and the initial pressures in both wetting and non-wetting fluid are zero. A pore pressure equal to 10 MPa is applied to the left model boundary (which is fully saturated). The following properties were used in the simulation: a = m ρ w = 1000 kg / m 3 ρ nw =1kg/m 3

215 FLUID FLOW IN JOINTS 2-69 k wj k nwj K w K nw = 83.3 Pa 1 sec 1 = 83.3 Pa 1 sec 1 = 2000 MPa =1MPa The state of the model after 1.4 seconds is illustrated in Figures Comparison of the position of the fluid front as a function of time, as calculated by UDEC with the analytical solution, is shown in Figure The agreement is very good. JOB TITLE : Filling Joint with Water UDEC (Version 5.00) LEGEND Oct :01:17 cycle time 1.396E+00 sec flow time = 1.396E+00 sec boundary plot domain pore pressures maximum pressure= 9.398E+06 each line thick = 1.880E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.30 Wetting fluid pressure in the joint after 1.43 s

216 2-70 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE : Filling Joint with Water UDEC (Version 5.00) LEGEND Oct :01:17 cycle time 1.396E+00 sec flow time = 1.396E+00 sec boundary plot domain saturation maximum saturat.= 1.000E+00 each line thick = 2.000E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.31 Saturation of the joint after 1.43 s JOB TITLE : Filling Joint with Water UDEC (Version 5.00) (e+007) 1.00 LEGEND 5-Oct :01:17 cycle time 1.396E+00 sec flow time = 1.396E+00 sec history plot Y-axis: 1 - domain pressure 2 - domain pressure 3 - domain pressure 4 - domain pressure 5 - domain pressure 6 - domain pressure 7 - domain pressure 8 - domain pressure 9 - domain pressure X-axis: Time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.32 Wetting fluid pressure histories at the points along the joint

217 FLUID FLOW IN JOINTS 2-71 JOB TITLE : Filling Joint with Water UDEC (Version 5.00) LEGEND 5-Oct :01:17 cycle time 1.396E+00 sec flow time = 1.396E+00 sec history plot Y-axis: 13 - domain saturation 14 - domain saturation 15 - domain saturation 16 - domain saturation 17 - domain saturation 18 - domain saturation 19 - domain saturation 20 - domain saturation 21 - domain saturation X-axis: Time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.33 Saturation histories at the points along the joint JOB TITLE : Filling Joint with Water (e-001) UDEC (Version 5.00) 4.00 LEGEND 5-Oct :01:17 cycle time 1.396E+00 sec flow time = 1.396E+00 sec table plot 3.23E-02<tab 3> 3.75E-01 X X X 0.00E+00<tab 4> 3.87E-01 Vs. 0.00E+00<X value> 9.00E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-001) Figure 2.34 Location of fluid front (distance from the left boundary in meters) as a function of time (seconds). Comparison of the UDEC solution (crosses) with analytical solution (line).

218 2-72 Special Features Structures/Fluid Flow/Thermal/Dynamics Example 2.10 Injection of water in gas-filled joint new ;file: filljoint.dat ; ; Filling of joint with fluid ; No capillary pressure ; config fluid set flow clear compressible twophase on round 2E-3 edge 4E-3 block -1,-1-1,1 1,1 1,-1 crack (-1,0) (1,0) gen edge 0.1 group zone block zone model elastic density 2.5E3 bulk 1E10 shear 5E9 range group block group joint joint joint model area jks 1E11 jkn 1E11 jperm 83.3 nwjperm 83.3 ares 1E-5 & azero 1E-5 range group joint ; new contact default set jcondf joint model area jks=1e11 jkn=1e11 jperm=83.3 nwjperm=83.3 & ares=1e-5 azero=1e-5 change dmat 1 prop dmat 1 acap bcap 2 gcap 0 fluid density= fluid bulkw=2.0e9 fluid nwdens=1.0 fluid nwbulk= initial sat 0.01 boundary xvelocity 0 range -1.01, ,-0.99 boundary yvelocity 0 range -1.01, ,-0.99 boundary yvelocity 0 range -1.01, ,1.01 boundary xvelocity 0 range -1.01, ,1.01 boundary pp 1.0E7 range -1.01, ,1.01 boundary nwpp 1.0E7 range -1.01, ,1.01 boundary sat 1.0 range -1.01, ,1.01 boundary pp 0.0 range 0.99, ,1.01 boundary nwpp 0.0 range 0.99, ,1.01 set gravity= history pp -0.97,0.0 history pp -0.94,0.0 history pp -0.91,0.0 history pp -0.88,0.0 history pp -0.84,0.0

219 FLUID FLOW IN JOINTS 2-73 history pp -0.81,0.0 history pp -0.78,0.0 history pp -0.75,0.0 history pp -0.72,0.0 history pp -0.69,0.0 history pp -0.66,0.0 history pp -0.62,0.0 history sat -0.97,0.0 history sat -0.94,0.0 history sat -0.91,0.0 history sat -0.88,0.0 history sat -0.84,0.0 history sat -0.81,0.0 history sat -0.78,0.0 history sat -0.75,0.0 history sat -0.72,0.0 history sat -0.69,0.0 history sat -0.66,0.0 history sat -0.62,0.0 save fj1.sav ; set mech=off cycle save fj2.sav ; def _analytic array xd(12) xd(1) = -0.97,0 xd(2) = -0.94,0 xd(3) = -0.91,0 xd(4) = -0.88,0 xd(5) = -0.84,0 xd(6) = -0.81,0 xd(7) = -0.78,0 xd(8) = -0.75,0 xd(9) = -0.72,0 xd(10) = -0.69,0 xd(11) = -0.66,0 xd(12) = -0.62,0 p_bar = i = 1 table(4,0.0) = 0.0 loop j (1,100) _tsat = 0.009*j _xsat =sqrt(2.0*p_bar*_tsat) table(4,_tsat) = _xsat

220 2-74 Special Features Structures/Fluid Flow/Thermal/Dynamics end_loop loop i (1,12) ih = i+12 command table 2 delete hist write ih table 2 end_command _index = 0 _s = 0 loop while _index = 0 _s = _s+1 _sat = ytable(2,_s) if _sat >= 0.5 then _index = _s _tsat = xtable(2,_s) _idom = d_near(xd(i),0.0) _xsat = 1.0+d_x(_idom) endif end_loop table(3,_tsat) = _xsat end_loop end _analytic save fj3.sav ;plot name: pp plot hold boundary pp white ;plot name: saturation plot hold boundary saturation white ;plot name: sat histrory plot hold history 13 line 14 line 15 line 16 line 17 line 18 line 19 line & 20 line 21 line 22 line 23 line 24 line vs time ;plot name: pp history plot hold history 1 line 2 line 3 line 4 line 5 line 6 line 7 line 8 & line 9 line 10 line 11 line 12 line vs time ;plot name: fluid front plot hold table 3 cross 4 line skip 1

221 FLUID FLOW IN JOINTS Filling of a Horizontal Joint by the Capillary Forces Only Filling of an initially air-filled joint with water driven by only capillary forces may be solved by UDEC. The material properties are the same as those described in Section 2.5.7, with the exception that aperture, a, is now m. The capillary pressure curve is defined by the following parameters: a 0 = m γ =10m β =2 The initial saturation inside the joint is 1%. The non-wetting fluid pressure inside the joint is initialized to zero, which implies that initial pressure in the wetting fluid is 10 MPa. Boundary conditions on the left boundary of the model were zero pressure of both wetting and non-wetting fluid, and full saturation (i.e., s w = 100%). During the simulation, the fluid from the left boundary is sucked inside the model by the capillary forces. The state of the model after.014 seconds is shown in Figures JOB TITLE : Joint Filling by Capillary Forces UDEC (Version 5.00) LEGEND Oct :56:19 cycle time 1.396E-02 sec flow time = 1.396E-02 sec boundary plot neg. domain pore pressures minimum pressure=-1.000e+07 each line thick = 2.000E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.35 Wetting fluid pressures along the joint after.014 s

222 2-76 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE : Joint Filling by Capillary Forces UDEC (Version 5.00) LEGEND Oct :56:19 cycle time 1.396E-02 sec flow time = 1.396E-02 sec boundary plot domain saturation maximum saturat.= 5.417E-01 each line thick = 1.083E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.36 Saturation along the joint after.014 s JOB TITLE : Joint Filling by Capillary Forces UDEC (Version 5.00) (e+007) 0.00 LEGEND 5-Oct :56:19 cycle time 1.396E-02 sec flow time = 1.396E-02 sec history plot Y-axis: 1 - domain pressure 2 - domain pressure 3 - domain pressure 4 - domain pressure 5 - domain pressure 6 - domain pressure 7 - domain pressure 8 - domain pressure 9 - domain pressure X-axis: Time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-002) Figure 2.37 Wetting fluid pressure histories at the points along the joint

223 FLUID FLOW IN JOINTS 2-77 JOB TITLE : Joint Filling by Capillary Forces UDEC (Version 5.00) (e-001) 6.00 LEGEND 5-Oct :56:19 cycle time 1.396E-02 sec flow time = 1.396E-02 sec history plot Y-axis: 13 - domain saturation 14 - domain saturation 15 - domain saturation 16 - domain saturation 17 - domain saturation 18 - domain saturation 19 - domain saturation 20 - domain saturation 21 - domain saturation X-axis: Time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-002) Figure 2.38 Saturation histories at the points along the joint

224 2-78 Special Features Structures/Fluid Flow/Thermal/Dynamics Example 2.11 Filling of the joint with water due to capillary forces only ; ; Filling of joint with fluid ; caused by capillary forces only ; config fluid set flow clear compressible twophase on round 2E-3 edge 4E-3 block -1,-1-1,1 1,1 1,-1 crack (-1,0) (1,0) gen edge 0.1 group zone block zone model elastic density 2.5E3 bulk 1E10 shear 5E9 range group block group joint joint joint model area jks 1E11 jkn 1E11 jperm 83.3 nwjperm 83.3 ares 1E-4 & azero 1E-4 range group joint ; new contact default set jcondf joint model area jks=1e11 jkn=1e11 jperm=83.3 nwjperm=83.3 & ares=1e-4 azero=1e-4 fluid density= fluid bulkw=2.0e9 fluid nwdens=1.0 fluid nwbulk= change dmat 1 property dmat 1 acap 1E-4 bcap 2 gcap 10 initial sat 0.01 initial nwpp 0.0 boundary xvelocity 0 range -1.01, ,-0.99 boundary yvelocity 0 range -1.01, ,-0.99 boundary xvelocity 0 range -1.01, ,1.01 boundary yvelocity 0 range -1.01, ,1.01 boundary pp 0.0 range -1.01, ,1.01 boundary nwpp 0.0 range -1.01, ,1.01 boundary sat 1.0 range -1.01, ,1.01 boundary pp -1.0E7 range 0.99, ,1.01 boundary nwpp 0.0 range 0.99, ,1.01 boundary sat 0.01 range 0.99, ,1.01 set gravity= hist pp -0.97,0 hist pp -0.94,0 hist pp -0.91,0 hist pp -0.88,0 hist pp -0.84,0

225 FLUID FLOW IN JOINTS 2-79 hist pp -0.81,0 hist pp -0.78,0 hist pp -0.75,0 hist pp -0.72,0 hist pp -0.69,0 hist pp -0.66,0 hist pp -0.62,0 hist sat -0.97,0 hist sat -0.94,0 hist sat -0.91,0 hist sat -0.88,0 hist sat -0.84,0 hist sat -0.81,0 hist sat -0.78,0 hist sat -0.75,0 hist sat -0.72,0 hist sat -0.69,0 hist sat -0.66,0 hist sat -0.62,0 save cfj1.sav ; set mech=off cycle save cfj2.sav ; ;plot name: pp plot hold boundary -pp int white ;plot name: saturation plot hold boundary saturation white ;plot name: pp history plot hold history 1 line 2 line 3 line 4 line 5 line 6 line 7 line 8 & line 9 line 10 line 11 line 12 line vs time ;plot name: sat history plot hold history 13 line 14 line 15 line 16 line 17 line 18 line 19 line & 20 line 21 line 22 line 23 line 24 line vs time

226 2-80 Special Features Structures/Fluid Flow/Thermal/Dynamics Containment of Gas Inside a Cavity Single Horizontal Joint A square opening, 0.4 m 0.4 m in cross section, is pressurized by the gas to 10 MPa. The opening intersects a joint that is fully saturated with water at atmospheric pressure. The geometry of the UDEC model representing the stated problem is shown in Figure The capillary forces are neglected in this problem. As the simulation is started, the non-wetting fluid enters the joint pushing the water outside (see Figures 2.40 and 2.41). The simulation of two-phase flow is fully coupled with mechanical deformation. The displacements of the model are shown in Figure Histories of the pressure of non-wetting fluid and saturation along the joints are shown in Figures 2.43 and JOB TITLE :. UDEC (Version 5.00) LEGEND :40:31 Wed Mar cycle 0 time 0.000E+00 sec flow time = 0.000E+00 sec block plot Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.39 Geometry of the model

227 FLUID FLOW IN JOINTS 2-81 JOB TITLE : Gas Containment UDEC (Version 5.00) LEGEND Oct :43:18 cycle time 3.623E-01 sec flow time = 1.279E-04 sec boundary plot domain pore pressures maximum pressure= 1.000E+07 each line thick = 2.000E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.40 Wetting fluid pressure along the joint after s JOB TITLE : Gas Containment UDEC (Version 5.00) LEGEND Oct :43:18 cycle time 3.623E-01 sec flow time = 1.279E-04 sec boundary plot domain saturation maximum saturat.= 1.000E+00 each line thick = 2.000E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.41 Saturation along the joint after s

228 2-82 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE : Gas Containment UDEC (Version 5.00) LEGEND Oct :43:18 cycle time 3.623E-01 sec flow time = 1.279E-04 sec block plot displacement vectors maximum = 5.008E E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2.42 Model displacements after s JOB TITLE : Gas Containment UDEC (Version 5.00) (e+007) 1.00 LEGEND 5-Oct :43:18 cycle time 3.623E-01 sec flow time = 1.279E-04 sec history plot Y-axis: 16 - non-wetting fluid po 17 - non-wetting fluid po 18 - non-wetting fluid po 19 - non-wetting fluid po 20 - non-wetting fluid po 21 - non-wetting fluid po 22 - non-wetting fluid po X-axis: 15 - fluid flow time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-004) Figure 2.43 Non-wetting fluid pressure histories at the points along the joint

229 FLUID FLOW IN JOINTS 2-83 JOB TITLE : Gas Containment UDEC (Version 5.00) 1.10 LEGEND 5-Oct :43:18 cycle time 3.623E-01 sec flow time = 1.279E-04 sec history plot Y-axis: 8 - domain saturation 9 - domain saturation 10 - domain saturation 11 - domain saturation 12 - domain saturation 13 - domain saturation 14 - domain saturation X-axis: 15 - fluid flow time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-004) Figure 2.44 Saturation at the points along the joint

230 2-84 Special Features Structures/Fluid Flow/Thermal/Dynamics Example 2.12 Containment of gas inside a cavity single joint config fluid set flow clear compressible twophase on round 2E-3 edge 4E-3 block -1,-1-1,1 1,1 1,-1 crack (-1,0) (1,0) crack (0.2,-0.2) (0.2,0.2) crack (0.2,0.2) (-0.2,0.2) crack (-0.2,0.2) (-0.2,-0.2) crack (-0.2,-0.2) (0.2,-0.2) delete range -0.2, ,0.2 gen quad 0.1 group zone block zone model elastic density 2.5E3 bulk 1E10 shear 5E9 range group block group joint joint joint model area jks 1E11 jkn 1E11 jperm 83.3 nwjperm 83.3 ares 1E-4 & azero 1E-3 range group joint ; new contact default set jcondf joint model area jks=1e11 jkn=1e11 jperm=83.3 nwjperm=83.3 & ares=1e-4 azero=1e-3 fluid density= fluid bulkw=2.0e9 fluid nwdens=1.0 fluid nwbulk= change dmat 1 property dmat 1 acap 1E-4 bcap 2 gcap 0 initial sat 1.0 initial sat 0.0 range -0.2, ,0.2 initial pp 0.0 pfix ppressure 1E7 range -0.2, ,0.2 pfix nwppressure 1E7 range -0.2, ,0.2 insitu stress -1.0E7,0.0,-1.0E7 boundary xvelocity 0 boundary yvelocity 0 boundary pp 0.0 range -1.01, ,1.01 boundary pp 0.0 range 0.99, ,1.01 boundary nwpp 0.0 range -1.01, ,1.01 boundary nwpp 0.0 range 0.99, ,1.01 history pp -0.24,0.0 history pp -0.28,0.0 history pp -0.32,0.0 history pp -0.36,0.0 history pp -0.4,0.0

231 FLUID FLOW IN JOINTS 2-85 history pp -0.44,0.0 history pp -0.48,0.0 history saturation -0.24,0.0 history saturation -0.28,0.0 history saturation -0.32,0.0 history saturation -0.36,0.0 history saturation -0.4,0.0 history saturation -0.44,0.0 history saturation -0.48,0.0 history flowtime history nwpp -0.24,0.0 history nwpp -0.28,0.0 history nwpp -0.32,0.0 history nwpp -0.36,0.0 history nwpp -0.4,0.0 history nwpp -0.44,0.0 history nwpp -0.48,0.0 set gravity= set capratio=10.0 save gas1.sav : cycle save gas2.sav ;*** plots **** ;plot name: pp plot hold boundary pp white ;plot name: Plot 1 plot hold boundary saturation white ;plot name: disp plot hold block displacement white ;plot name: pp hist plot hold history vs 15 ;plot name: Plot 2 plot hold history 8 line 9 line 10 line 11 line 12 line 13 line 14 line & vs 15

232 2-86 Special Features Structures/Fluid Flow/Thermal/Dynamics Containment of Gas Inside a Cavern Jointed Rock Mass Containment of a gas inside a cavern in a jointed rock mass is modeled using UDEC. The geometry of the model is shown in Figure The same material properties as in the example from Section are used in this problem. The size of the cavern is 4.2 m 4.2 m. The joints in the rock mass are initially fully saturated with water. The pressure in the water is equal to atmospheric pressure. The non-wetting fluid pressure inside the cavern is equal to 10 MPa. The state of the model, after 0.25 seconds, as gas pushes out the water is illustrated in Figures 2.46 and JOB TITLE : Gas Containment in Jointed Rock (*10^1) UDEC (Version 5.00) LEGEND Oct :43:45 cycle 0 time 0.000E+00 sec flow time = 0.000E+00 sec block plot Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 2.45 Geometry of the model

233 FLUID FLOW IN JOINTS 2-87 JOB TITLE : Gas Containment in Jointed Rock (*10^1) UDEC (Version 5.00) LEGEND Oct :17:44 cycle time 4.418E-01 sec flow time = 2.500E-01 sec boundary plot domain non-wetting fl. pres maximum pressure= 1.000E+07 each line thick = 2.000E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 2.46 Non-wetting fluid pressures after 0.25 s JOB TITLE : Gas Containment in Jointed Rock (*10^1) UDEC (Version 5.00) LEGEND Oct :17:44 cycle time 4.418E-01 sec flow time = 2.500E-01 sec boundary plot domain saturation maximum saturat.= 1.000E+00 each line thick = 2.000E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 2.47 Saturation of the joints in the model after 0.25 s

234 2-88 Special Features Structures/Fluid Flow/Thermal/Dynamics Example 2.13 Containment of gas inside a cavity jointed rock mass config fluid set flow clear compressible twophase on round 2E-2 edge 4E-2 block -10,-10-10,10 10,10 10,-10 jset angle 45 spacing 1 origin 0,0 jset angle 135 spacing 1 origin 0,0 crack (2.12,-2.12) (2.12,2.12) crack (2.12,2.12) (-2.12,2.12) crack (-2.12,2.12) (-2.12,-2.12) crack (-2.12,-2.12) (2.12,-2.12) delete range -2.12, ,2.12 delete area 0.1 gen edge 1.0 group zone block zone model elastic density 2.5E3 bulk 1E10 shear 5E9 range group block group joint joint joint model area jks 1E11 jkn 1E11 jcohesion 1E10 jtension 1E10 & jperm 83.3 nwjperm 83.3 ares 1E-4 azero 1E-4 range group joint ; new contact default set jcondf joint model area jks=1e11 jkn=1e11 jcohesion=1e10 & jtension=1e10 jperm=83.3 nwjperm=83.3 ares=1e-4 azero=1e-4 fluid density= fluid bulkw=2.0e9 fluid nwdens=1.0 fluid nwbulk= change dmat 1 property dmat 1 acap 1E-4 bcap 2 gcap 0 initial sat 1.0 initial sat 0.0 range -2.12, ,2.12 initial pp 0.0 pfix ppressure 1E7 range -2.12, ,2.12 pfix nwppressure 1E7 range -2.12, ,2.12 insitu stress -1.0E7,0.0,-1.0E7 boundary xvelocity 0 boundary yvelocity 0 boundary pp 0.0 range -10.1, ,10.1 boundary nwpp 0.0 range -10.1, ,10.1 boundary pp 0.0 range 9.8, ,10.1 boundary nwpp 0.0 range 9.8, ,10.1 boundary pp 0.0 range -10.1, ,-9.8 boundary nwpp 0.0 range -10.1, ,-9.8 boundary pp 0.0 range -10.1, ,10.1

235 FLUID FLOW IN JOINTS 2-89 boundary nwpp 0.0 range -10.1, ,10.1 set gravity= save joint1.sav ; history saturation 2.2,0.0 history saturation 2.5,0.0 cycle save joint2.sav ; ;*** plots **** ;plot name: Plot 1 plot hold boundary nwpp white ;plot name: saturation plot hold boundary saturation white

236 2-90 Special Features Structures/Fluid Flow/Thermal/Dynamics Thermal-Mechanical-Fluid Flow Example This simple example shows the effect of a heat source on a saturated system initially at equilibrium. Figure 2.48 shows the jointed system with two orthogonal joint sets spaced 5 m apart. The properties are as follows. Elastic Blocks mass density 2500 kg/m 3 bulk modulus 6666 MPa shear modulus 4000 MPa thermal conductivity 5 W/m C thermal expansion coefficient 10 5 (1/C ) specific heat 900 J/kg C Coulomb Joints normal stiffness shear stiffness friction angle cohesion residual aperture aperture at zero stress 50,000 MPa/m 10,000 MPa/m MPa 20 μm 68 μm The fluid density is supposed to vary from 1000 kg/m 3 at 0 C (initial temperature) to 800 kg/m 3 at 100 C.

237 FLUID FLOW IN JOINTS 2-91 JOB TITLE : Thermal-Mechanical-Fluid Flow Test (*10^1) UDEC (Version 5.00) LEGEND Oct :17:47 cycle time 4.008E+00 sec thermal time = 6.000E+03 sec flow time = 6.000E+03 sec block plot History Locations Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 2.48 UDEC model for thermal-mechanical-fluid flow example The model is subjected to the gravitational stress state (units in Pa) σ xx = ( )y σ yy = ( )y The bottom boundary is fixed in the y-direction, and the side boundaries are fixed in the x-direction. The bottom and side boundaries are impermeable, and the model is fully saturated, with the water table 10 m above the top of the model. Adiabatic conditions are specified for the bottom and side boundaries, and a constant heat source of 100 kw/m is applied along the top boundary for 6000 seconds; thereafter, the boundary is adiabatic. The problem is solved with both the steady-state and transient (incompressible fluid) flow modes. Figures 2.49 and 2.50 show the fluid pressure history at three points in the model (see Figure 2.48). When the rock is heated, large thermal stresses are generated and compress the joints. This causes a transient increase of water pressures. However, once these extra pressures have dissipated, the only remaining effect on the fluid is the decrease in its density. Thus, fluid pressures stabilize at a level lower than their initial values. The steady-state flow algorithm directly reproduces the long-term behavior without going through the early time increase, whereas the transient flow algorithm shows the complete sequence. (Compare Figures 2.49 and 2.50.) The data files for these two cases are given in Examples 2.14 and 2.15.

238 2-92 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE : Thermal-Mechanical-Fluid Flow Test UDEC (Version 5.00) (e+005) 7.00 LEGEND 5-Oct :17:58 cycle time 5.158E+00 sec thermal time = 1.200E+05 sec flow time = 1.200E+05 sec history plot Y-axis: 3 - domain pressure 4 - domain pressure 5 - domain pressure X-axis: 10 - thermal time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+005) Figure 2.49 Fluid pressure versus thermal time histories for transient flow analysis JOB TITLE : Thermal-Mechanical-Fluid Flow Test UDEC (Version 5.00) (e+005) 7.00 LEGEND 5-Oct :18:40 cycle time 3.352E+00 sec thermal time = 2.700E+04 sec flow time = 3.352E+00 sec history plot Y-axis: 3 - domain pressure 4 - domain pressure 5 - domain pressure X-axis: 10 - thermal time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+004) Figure 2.50 Fluid pressure versus thermal time histories for steady-state flow analysis

239 FLUID FLOW IN JOINTS 2-93 Example 2.14 Thermo-mechanical-fluid flow with incompressible transient flow config thermal fluid set flow clear incompressible on round 0.1 edge 0.2 block 0,0 0,40 20,40 20,0 jset angle 90 spacing 5 origin 0,0 jset angle 0 spacing 5 origin 0,0 gen edge 10.0 ; elastic block (E=100000bars, nu=.25) group zone block zone model elastic density 2.5E3 bulk E9 shear 4E9 cond 5 specheat & 900 thexp 1E-5 range group block ; Coulomb joints group joint joint joint model area jks 1E10 jkn 5E10 jfriction 30 jcohesion 1E7 jperm 83 & ares azero 6.8E-5 range group joint ; new contact default set jcondf joint model area jks=1e10 jkn=5e10 jfriction=30 jcohesion=1e7 & jperm=83 ares=2e-5 azero=6.8e-5 ; water fluid density= fluid dtable=1 table 1 delete table 1 0 1E ; boundary conditions bou stress -10e5 0-20e5 pp 10e4 range bou xvel=0 imperm range bou xvel=0 imperm range bou yvel=0 imperm range ; gravity and initial stresses set gravity=0-10 insitu stres -15e5 0-30e5 ygrad.125e5 0.25e5 ywtable 50. his flowtime hist pp 5,7.5 hist pp 5,20 hist pp 5,30 hist ydisp 5,40 history ncyc 1 history ntcyc 1 set capratio=50.0 ; incompressible flow parameters set dtflow=60.0 set voltol=0.001

240 2-94 Special Features Structures/Fluid Flow/Thermal/Dynamics ; equilibrate without heat cycle 3000 save thm_inc1.sav ; reset hist time set ftime=0 history ncyc 1 history ntcyc 1 his flowtime hist pp 5,7.5 hist pp 5,20 hist pp 5,30 hist ydisp 5,40 history temperature 10.0,40.0 history temperature 10.0,35.0 history temperature 10.0,30.0 history thtime ; heat source at top thapp flux 1e5 0 range set nmech=1 set ntherm=1 set thdt=60.0 run temp 100 step 100 save thm_inc2.sav ; stop heat source and let pressures stabilize thapp flux -1e5 0 range run step 1900 save thm_inc3.sav ; ;*** plot commands **** ;plot name: unbal force plot hold history 1 line ;plot name: pressure hist plot hold history vs 10 ;plot name: ydisp hist plot hold history 6 line ret

241 FLUID FLOW IN JOINTS 2-95 Example 2.15 Thermal-mechanical-fluid flow assuming steady-state flow config thermal fluid set flow steady on round 0.1 edge 0.2 block 0,0 0,40 20,40 20,0 jset angle 90 spacing 5 origin 0,0 jset angle 0 spacing 5 origin 0,0 gen edge 10.0 ; elastic block (E=100000bars, nu=.25) group zone block zone model elastic density 2.5E3 bulk E9 shear 4E9 cond 5 specheat & 900 thexp 1E-5 range group block ; Coulomb joints group joint joint joint model area jks 1E10 jkn 5E10 jfriction 30 jcohesion 1E7 jperm 83 & ares azero 6.8E-5 range group joint ; new contact default set jcondf joint model area jks=1e10 jkn=5e10 jfriction=30 jcohesion=1e7 & jperm=83 ares=2e-5 azero=6.8e-5 ; water fluid density= fluid dtable=1 table 1 delete table 1 0 1E ; boundary conditions bou stress -10e5 0-20e5 pp 10e4 range bou xvel=0 imperm range bou xvel=0 imperm range bou yvel=0 imperm range ; gravity and initial stresses set gravity=0-10 insitu stres -15e5 0-30e5 ygrad.125e5 0.25e5 ywtable 50. his flowtime hist pp 5,7.5 hist pp 5,20 hist pp 5,30 hist ydisp 5,40 history ncyc 1 history ntcyc 1 set capratio=50.0 ; equilibrate without heat solve ratio 1e-6 save thm_ss1.sav

242 2-96 Special Features Structures/Fluid Flow/Thermal/Dynamics ; reset hist time set ftime=0 history ncyc 1 history ntcyc 1 his flowtime hist pp 5,7.5 hist pp 5,20 hist pp 5,30 hist ydisp 5,40 history temperature 10.0,40.0 history temperature 10.0,35.0 history temperature 10.0,30.0 hist thtime ; heat source at top thapp flux 1e5 0 range set nmech=30 set ntherm=1 set thdt=60.0 run temp 100 step 100 save thm_ss2.sav ; stop heat source and let pressures stabilize thapp flux -1e5 0 range run step 350 save thm_ss3.sav ;*** plot commands **** ;plot name: unbal force plot hold history 1 line ;plot name: pressure hist plot hold history vs 10 ;plot name: ydisp hist plot hold history 6 line ret

243 FLUID FLOW IN JOINTS References Abdallah, G., et al. Thermal Convection of Fluid in Fractures Media, Int. J. Rock Mech. Min Sci. & Geomech. Abstr., 32(5), (1995). Barton, N., S. Bandis and K. Bakhtar. Strength, Deformation and Conductivity Coupling of Rock Joints, Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 22(3), (1985). Carslaw, H. S., and J. C. Jaeger. Conduction of Heat in Solids, 2nd Ed. Oxford: Clarendon Press (1959). Chan, D. Y. C., B. D. Hughes and L. Paterson. Transient gas flow around boreholes, Transport in Porous Media, 10, (1993). Detournay, E., and A. H.-D. Cheng. Fundamentals of Poroelasticity, in Comprehensive Rock Engineering, Vol. 2, pp J. Hudson et al., eds. London: Pergamon Press (1993). Hardy, M. P., and M. J. Asgian. Fracture Reopening during Hydraulic Fracturing Stress Determinations, Int. J. Rock Mech. Min Sci. & Geomech. Abstr., 26(6), (1989). Harr, M. E. Groundwater and Seepage. New York: McGraw-Hill Book Company (1962). Itasca Consulting Group Inc. UDEC (Universal Distinct Element Code), Version 5.0. Minneapolis: ICG (2011). Louis, C. A Study of Groundwater Flow in Jointed Rock and Its Influence on the Stability of Rock Masses, Imperial College, Rock Mech. Research Report No. 10 (1969). Pruess, K., and Y. W. Tsang. On Two-Phase Relative Permeability and Capillary Pressure of Rough-Walled Rock Fractures, Water Resources Research, 26(9), (September 1990). Voller, V. R., S. Peng and Y. F. Chen. Numerical Solution of Transient, Free Surface Problems in Porous Media, Int. J. Num. Meth. Eng., 39, (1996). Wilkinson, W. L. Non-Newtonian Fluids: Fluid Mechanics, Mixing and Heat Transfer. London: Pergamon Press (1960). Witherspoon, P. A., et al. Validity of Cubic Law for Fluid Flow in a Deformable Rock Fracture, Water Resources Research, 16(6), (1980).

244 2-98 Special Features Structures/Fluid Flow/Thermal/Dynamics

245 THERMAL ANALYSIS THERMAL ANALYSIS 3.1 Introduction UDEC allows simulation of transient heat conduction in materials, and the development of thermally induced displacements and stresses. This includes the following specific features. 1. Heat transfer is modeled as conduction either isotropic or anisotropic, depending on the user s choice of material properties. 2. Several different thermal boundary conditions may be imposed. 3. Any of the mechanical block models may be used with the thermal model. 4. Heat sources may be inserted into the material as volume sources. The sources may be made to decay exponentially with time. 5. Both implicit and explicit calculations schemes are available, and the user can switch from one to the other at any time during a run. 6. The thermal analysis provides one-way coupling to the mechanical stress calculation through the thermal expansion coefficient. 7. The thermal analysis provides one-way coupling to the calculation for fluid flow in joints through the temperature dependency of fluid density and joint permeability. This section contains a description of the thermal formulation (Section 3.2). Recommendations for solving thermal and thermal-mechanical problems are also provided (Section 3.3). The UDEC input commands for thermal analysis (Section 3.4), and the system of units for thermal analysis (Section 3.5) are given. Finally, several verification problems (Section 3.6) are described.* Refer to these examples as a guide for creating UDEC models for thermal analysis and coupled thermal-stress analysis. See Section 2 for a description of coupled thermal-fluid flow analysis. * The data files in this section are stored in the directory ITASCA\UDEC500\Datafiles\Thermal with the extension.dat. A project file is also provided for each example. In order to run an example and compare the results to plots in this section, open a project file in the GIIC by clicking on the File / Open Project menu item and selecting the project file name (with extension.prj ). Click on the Project Options icon at the top of the Project Tree Record, select Rebuild unsaved states, and the example data file will be run, and plots created.

246 3-2 Special Features Structures/Fluid Flow/Thermal/Dynamics 3.2 Formulation Basic Equations The basic equation of conductive heat transfer is Fourier s law, which can be written in one dimension as Q i = k ij T x j (3.1) where Q i = flux in the i-direction (W/m 2 ); k ij = thermal conductivity tensor (W/m C); and T = temperature. Also, for any mass, the change in temperature can be written as T t = Q net C p M (3.2) where Qnet = net heat flow into mass (M); C p = specific heat (J/kg C); and M = mass (kg). These two equations are the basis of the thermal version of UDEC. For two-dimensional heat transfer, Eq. (3.2) can be written as T t = 1 C p ρ [ Qx x + Q y y ] (3.3) where ρ is the mass density. Combining this with Eq. (3.1), T t = 1 C p ρ x [ k x T x ] + y [ k y T y ] (3.4) = 1 ρc p [ k x 2 T x 2 + k y 2 T y 2 ]

247 THERMAL ANALYSIS 3-3 if k x and k y are constant. This is called the diffusion equation. Temperature changes cause stress changes according to the equation σ ij = δ ij 3K * α T (3.5) where σ ij = change in stress ij; δ ij = Kronecker delta (δ ij = 1 for i = j and 0 for i j); K * = K (for plane strain); =6KG/(3K + 4G) for plane stress, where K is bulk modulus and G is shear modulus; α = linear thermal expansion coefficient; and T = temperature change. The mechanical changes can also cause temperature changes as energy is dissipated in the system. This effect is neglected because it is usually negligible Diffusion Equation Explicit Algorithm UDEC discretizes fully deformable blocks into triangular zones which are also used for the thermal analysis. At each timestep, Eqs. (3.1) and (3.2) are solved numerically, using the following scheme. 1. In each triangle, ( T / x and T/ y) are approximated using the equation T x i = 1 A 3 Tn i ds (3.6) = 1 A m=1 T m ɛ ij x m j where A = area of the triangle; n i = i th component of outward normal; T m = average temperature on side m; xj m = difference in x j between ends of side m; and ɛ ij = two-dimensional permutation tensor. ( ) 0 1 ɛ ij = 1 0

248 3-4 Special Features Structures/Fluid Flow/Thermal/Dynamics The heat flow into each gridpoint of the triangle is calculated from F i = A j Q i (3.7) where A j is the width of the line perpendicular to the component Q i, as shown in Figure 3.1. F total = F x + F y (3.8) = A y Q x + A x Q y k k A y A x Figure 3.1 Heat flow into gridpoint k 2. For each gridpoint, T = Q net t (3.9) C p M where Qnet is the sum of F totals from all zones affecting gridpoint i Stability and Accuracy of the Explicit Scheme For the explicit scheme, t is limited by numerical stability considerations. The critical timestep for stability, assuming x = the smallest zone dimension in the model (see, for example, Karlekar and Desmond 1982), is ( x) 2 t 4κ [ 1 + h x ] (3.10) 2k where h = the convective heat transfer coefficient; and κ = the thermal diffusivity (k/ρc p for k x = k y = k).

249 THERMAL ANALYSIS 3-5 The accuracy of the explicit solution scheme is determined by the introduction of errors from several sources. A strict definition of error in the explicit formulation is not obtained, simply because error arises from the finite difference approximations used; the error also is affected by the zone discretization and timestep. The explicit solution introduces a mixed order of error in the diffusion equation. This is because a forward difference formulation is used in time, which is first-order accurate, and a central difference formulation is used in spatial coordinates, which is second-order accurate Diffusion Equation Implicit Thermal Logic The implicit thermal logic in UDEC uses the Crank-Nicholson method, and the set of equations is solved by an iterative scheme known as the Jacobi method. An implicit method is advantageous for solving linear problems, such as heat conduction with constant conductivity, because it allows the use of much larger timesteps than those permitted by an explicit method, particularly at later times in a problem, when temperatures are changing slowly. The usual one-dimensional, explicit finite-difference scheme for heat conduction can be written as ρc p k T i (t + T ) T i (t) t = T i+1(t) 2 T i (t) + T i 1 (t) ( x) 2 (3.11) An implicit method can be derived by replacing the right-hand side of Eq. (3.11) with the expression [ 1 Ti+1 (t + t) 2 T i (t + t) + T i 1 (t + t) 2 ( x) 2 + T ] i 1(t) 2 T i (t) + T i 1 (t) ( x) 2 This method, known as the Crank-Nicholson method, has the advantage that it is stable for all values of t, but it has the disadvantage of being implicit. This means that the temperature change at any point depends on the temperature change at other points. This can be seen by rewriting the implicit scheme as [ ρc p Ti+1 k t T i = T i+1 2 (T i T i) + T i T i 1 ( x) 2 ] (3.12) since T k (t + t) = T k (t) + T k. The implicit method requires that a set of equations be solved at each timestep for the values of T i. In matrix notation, the explicit method can be written as T = C T ] (3.13)

250 3-6 Special Features Structures/Fluid Flow/Thermal/Dynamics where C is a coefficient matrix; T is a vector of the temperatures; and T is a vector of the temperature change. The implicit scheme can be written as T = C ( T T ) (3.14) which can be rewritten as ( I 1 ) 2 C T = C T (3.15) where we need to solve for T at each timestep. The matrix (I 1 2 C) is diagonally dominant and sparse, because only neighboring points contribute nonzero values to C. Thus, this set of equations is efficiently solved by an iterative scheme. For ease of implementation as a simple extension of the explicit method, the Jacobi method is used. For the N N system Ax = b, this can be written for the n th iteration as x i (n + 1) = b i a ii N j=1 j 1 [ ] aij x j (n) a ii i = 1, 2,...N (3.16) That is, x i (n + 1) = 1 a ii [ b i N j=1 ] a ij x j (n) + x i (n)

251 THERMAL ANALYSIS 3-7 In our case, this becomes T i (n + 1) = C ii [ N j=1 N C ij T j (δ ij 1 ] 2 C ij ) T j (n) + j=1 T i (n) (3.17) = C ii [ N j=1 C ij T j N j=1 ] C ij T j (n) T i (n) + T i (n) This equation shows the analogy between the implicit scheme and the explicit scheme, which can be written as T i = N j=1 C ij T j (3.18) The amount of calculation required for each timestep is approximately n + 1 times that required for one timestep in the explicit scheme, where n is the number of iterations per timestep. This extra calculation can be more than offset by the much larger timestep permitted by the implicit method, which makes the implicit scheme advantageous when the temperature change is linear in time Stability and Accuracy of the Implicit Scheme As described previously, the implicit solution scheme has the advantage that it is unconditionally stable for all timesteps. However, the differencing scheme presented in Eq. (3.11) assumes that the temperature change is a linear function of time in a single timestep. Depending on the problem to be modeled, this assumption may lead to inaccurate results if temperature gradients are very high or are changing very rapidly (e.g., at early times in a simulation). The code uses a Jacobi iteration method to solve the system of equations at every timestep. From a strictly numerical perspective, convergence of the iteration is achieved if a ii > N j=1 j i a ij i = 1, 2,..., N (3.19) where a ij are the previously described coefficients of the solution matrix A.

252 3-8 Special Features Structures/Fluid Flow/Thermal/Dynamics The above condition simply means that it is possible to obtain a numerical solution to the system of equations but that the solution has no bearing on the accuracy with which the derived solution compares to the true solution. There is no explicit method for determination of convergence to the true solution as a function of timestep, since the convergence depends on many factors (including properties, grid dimensions and grading, and boundary conditions). In most cases, the critical timestep (from Eq. (3.10)) provides a lower-bound estimate for the implicit timestep. A trial-and-error procedure is required to set the timestep above this value. Typically, a thermal problem is set up and initialized using the explicit procedure Thermal-Stress Coupling The heat transfer may be coupled to thermal-stress calculations at any time during a transient simulation. The coupling occurs in one direction only (i.e., the temperature may result in stress changes, but mechanical changes in the body resulting from force application do not result in temperature change). This restriction is not believed to be of great significance here, since the energy changes for quasi-static mechanical problems is usually negligible. The stress change in a triangular zone is given by (from Eq. (3.5)) σ ij = δ ij 3K α T (3.20) This assumes a constant temperature in each triangular zone, which is interpolated from the surrounding gridpoints. This stress is added to the zone stress state prior to application of the constitutive law.

253 THERMAL ANALYSIS Solving Thermal-Only and Coupled-Thermal Problems UDEC has the ability to perform thermal analysis and coupled thermal-mechanical and thermal-fluid flow analysis. The form of the coupled thermal-mechanical interaction is described in Section The coupling of thermal analysis with fluid flow in joints is described in Section In all cases, the CONFIG command must be given with the thermal keyword before the BLOCK command is specified. The procedure and required commands to implement the thermal-only and thermal-mechanical analysis approach is described in the following sections. The application of the thermal analytic capability is illustrated by several verification problems in Section Thermal Analysis UDEC can perform both transient and steady-state thermal analysis. The thermal calculation is performed with the RUN command. In order to perform a thermal-only analysis, the linking to the mechanical calculation must be suppressed with the command SET nther=0. This is the default state. Both explicit and implicit solution methods are available for thermal analysis. By default, an explicit solution procedure is invoked with the RUN command. The thermal timestep is calculated from Eq. (3.10). A number of thermal steps can be specified with the RUN step command. Alternatively, a heating time limit, in seconds, can be specified with the RUN age command. The change in temperature during one thermal timestep is limited to 20, by default. The thermal calculation will stop if this limit is exceeded. The limit can be changed with the temperature keyword following the RUN command, or the temperature change can be reduced by reducing the thermal timestep with the RUN delt or SET thdt command. The thermal timestep is printed to the screen when RUN is given. The timestep can also be obtained with the PRINT info command. The implicit solution algorithm described in Section is implemented with the keyword implicit following the RUN command. The thermal timestep can then be adjusted with the keyword delt following the RUN command, or with the SET thdt command. It is permissible to change between implicit and explicit solution methods at any time during a run, using the RUN and RUN implicit commands. The explicit scheme is always used unless the keyword implicit is given. The advantage of an implicit method is that the timestep thdt is not restricted by numerical stability. There are three disadvantages: (1) extra memory is required to use this method; (2) a set of simultaneous equations must be solved at each timestep; and (3) larger timesteps may introduce inaccuracy. These disadvantages must be kept in mind when deciding which method to use. They are discussed below.

254 3-10 Special Features Structures/Fluid Flow/Thermal/Dynamics Memory Requirement If an attempt is made to use the implicit method for a problem when the UDEC memory is almost full (typically, when the PRINT mem command reports at least 95% full), an error message may be generated. The only way to avoid this is to run a smaller problem or use the explicit method. Solving a Set of Equations The set of equations to be solved at each timestep is solved iteratively. Each iteration of the solution takes about the same length of time as a single step of the explicit method. The number of iterations depends on the timestep chosen and the particular problem being solved, but is always at least 3. Thus, the implicit scheme only offers an advantage over the explicit scheme if the timestep is much larger than that which the explicit scheme would use. On the other hand, the iterative scheme does introduce some restriction on the timestep. In general, a timestep between 100 and 10,000 times that used by the explicit scheme is satisfactory. The program displays the iteration counter and a measure of convergence (the residual) to the left of the timestep counter while the implicit scheme is running. The user should check that the number of iterations being taken is such that the implicit scheme is indeed more efficient than the explicit scheme. If not, switch to the explicit scheme or change the timestep. This counter will also indicate whether the method is not converging. If the residual is increasing with successive iterations, the method is not converging, and a smaller timestep must be used. Inaccuracy due to Large Timesteps In the initial period of a solution, temperatures generally change much faster than later in the solution period. In addition, the implicit scheme uses more iterations when modeling rapid changes. It is appropriate, therefore, to use a smaller timestep or the explicit method, initially, and to then switch to the implicit method with a large timestep later in the solution period. Convergence of the solution generally occurs in fewer iterations at later timesteps. Selecting the Implicit Method From the preceding discussion, it can be seen that the implicit method is most efficient when used at late times in the solution, and only if the timestep can be increased significantly over the one used by the explicit scheme Thermal-Mechanical Analysis The thermal calculation can be combined with the mechanical calculation to perform a thermalmechanical analysis with UDEC. All the features of the thermal calculation (including transient and steady-state heat transfer, and thermal solution by either the explicit or implicit algorithm) are available in a thermal-mechanical calculation. The thermal-mechanical coupling is provided by the influence of temperature change on the volumetric change of a zone (see Eq. (3.20)). The linear thermal expansion coefficient is assigned via the keyword thexp given with the PROPERTY command. The thermal model applies to all zones in the UDEC model. If zones are made null mechanically, the thermal model automatically is made null as well. The thermal-mechanical coupling can be invoked for any of the built-in mechanical constitutive models for plane-strain analysis. Plane-stress analysis can only be performed with the elastic isotropic and strain-hardening/softening models.

255 THERMAL ANALYSIS 3-11 The most common way to use UDEC to solve thermomechanical problems is to come to initial mechanical equilibrium and then take thermal steps to a time of interest. Remember that transient thermal problems involve time (e.g., the solution may be required after 10 years of heating). At this point, the mechanical problem has not been solved, although temperatures have been calculated. Mechanical steps are then taken until equilibrium is reached. This process is illustrated in Figure 3.2: 1. SETUP configuration for thermal analysis (CONFIG thermal) define problem geometry define material models and properties define thermal models and properties set boundary conditions (thermal & mechanical) set initial conditions (thermal & mechanical) set any internal conditions, such as heat sources 2. STEP TO EQUILIBRATE MECHANICALLY (STEP or SOLVE) PERFORM ANY DESIRED ALTERATIONS such as excavations. STEP TO EQUILIBRATE MECHANICALLY. REPEAT steps 3 and 4 until "initial" mechanical state is reached for thermal analysis. 5. TAKE THERMAL TIMESTEPS until desired time is reached (RUN). 6. STEP TO EQUILIBRATE MECHANICALLY (STEP or SOLVE). REPEAT steps 5 and 6 until sufficient time has been simulated. REPEAT steps 3 to 6 as necessary. Figure 3.2 General solution procedure for thermal-mechanical analysis UDEC can be used in the usual way to model the excavation of material, change material properties and change boundary conditions. The mechanical logic (the standard UDEC program) is also used in the thermomechanical program to take snapshots of the mechanical state at appropriate intervals in the development of the transient thermal stresses. The SOLVE, STEP or CYCLE command is used to control the mechanical steps. The RUN command controls the thermal process. For a problem in which the number of thermal steps is small before mechanical stepping is needed, analyses require a sequence of many RUN and STEP commands, which can be cumbersome to create and run. Therefore, it is possible to use the RUN command to switch automatically to mechanical steps during a series of thermal steps, using the SET nther and SET nmech commands. SET nther specifies the increment of thermal steps at which mechanical steps are to be taken. If nther is not zero, the calculation will switch to mechanical stepping every

256 3-12 Special Features Structures/Fluid Flow/Thermal/Dynamics nther steps, or when the temperature change parameter (set with RUN temp) is exceeded. SET nmech specifies the maximum number of mechanical steps executed between thermal steps. The mechanical calculation sub-stepping will stop when either the maximum number of mechanical steps is reached or the maximum unbalanced force ratio becomes smaller than The default is nmech=500. A difficulty with thermal-mechanical analysis is that a large temperature increase may cause a large increase in unbalanced forces in the blocks. If the analysis being performed is linear-elastic, no temperature increase will be too great, and UDEC need only equilibrate when the simulation time is such that a solution is required. For nonlinear problems, it is necessary to experiment to obtain an acceptable temperature increase effect on unbalanced forces. This can be done with the following steps. 1. Save the mechanical equilibrium state reached by UDEC. This state may be restored later for additional analyses. 2. Plot the stresses and shear displacements. If the stresses are near yield, the thermal stresses caused by the temperature changes should not be large. If the stresses are far from yield, larger stresses can be tolerated. 3. Run thermal steps until a particular temperature increase is reported by the program (using a RUN temp command). 4. Cycle mechanically to attain equilibrium. 5. Again, plot the stresses and shear displacements. If the area where the stresses are at or near yield is not much larger than at step 2, and the shear displacements are not very different, the allowed temperature increase was acceptable. If the changes are judged to be too great, the run must be repeated with a smaller allowed temperature change. It is important to note that the same temperature increase is not necessarily acceptable for all times in a problem. While the system is far from yield, large temperature increases will be acceptable; near yield, only relatively small increases can be tolerated Heat Transfer across Joints Heat transfers across the joints between blocks without resistance, provided that the blocks are in contact. Contacts are created along block edges for deformable blocks when the blocks are divided into triangular zones for mechanical calculations. For thermal calculations, the same zoning is used, with the exception that the triangles are further subdivided where the block is in contact with a corner on another block (Figure 3.3). Rigid blocks are divided into triangles using the centroid as a common vertex of all the triangles, with the other vertices at the corner and at the contact with corners on other blocks (Figure 3.4).

257 THERMAL ANALYSIS 3-13 Figure 3.3 Subdivision of zones at contacts rigid block Figure 3.4 Typical zoning of rigid block If the schemes outlined above are used without modification, it would be possible for very narrow triangles (such as those shown in Figure 3.5) to be formed. This causes inaccuracy, and may also lead to extremely small thermal timesteps. To avoid this, it is important to not have blocks with small zones neighboring blocks with large zones or large rigid blocks. The thermal tolerance option (keyword tol) on the RUN command should also be used to force points such as A and B in Figure 3.5 to be treated as one for thermal calculations. The tolerance may need to be reduced for models that contain small blocks or fine zoning. The value for tol must be smaller than the smallest block or zone edge length if the gridpoints associated with the smallest edge are not to be combined for the thermal calculation. For this reason, the tolerance is reduced for the verification examples in Section 3.6.

258 3-14 Special Features Structures/Fluid Flow/Thermal/Dynamics A B Figure 3.5 Zones that may cause inaccuracy Thermal Boundary Locations When modeling an infinite region, it is necessary to truncate the UDEC grid far enough away from the region of interest that the boundaries do not affect the solution. To determine whether the boundaries are far enough away, follow these steps: 1. Let the boundary representing infinity be insulated (the default boundary condition). 2. Solve the problem. 3. Examine the temperature changes on the boundary. 4. If the temperature changes are small, it is safe to assume the boundary has a negligible effect. If the temperature changes are not small, the boundary is probably too close. To confirm this, or disprove it, rerun the problem with the boundary temperatures fixed at their initial values. If the results are significantly different, the boundary was too close.

259 THERMAL ANALYSIS Input Instructions for Thermal Analysis UDEC Commands The following commands are provided to run thermal problems. Note that several thermal commands are invoked by new keywords used with existing commands in the standard mechanical code. The command CONFIG thermal must be given before the BLOCK command whenever a thermal analysis is to be performed. A summary of the thermal commands is given in Table 3.1: Table 3.1 Summary of thermal commands set thermal mode CONFIG thermal thermal properties PROPERTY cond xcond, ycond specheat, thexp ktable FLUID dtable initialize temperatures INITEM TADD TFIX TFREE specify thermal THAPP convection, flux boundary conditions radiation, source thermal-only solution SET nther = 0 coupled thermal- SET nmech, nther, thdt mechanical solution RUN age, delt, noage temp, step, tol, implicit output options PRINT gridpoint temp hist prop thermal thermal HISTORY temperature thtime PLOT boundary thermal, hist temperature, tfix RESET hist CONFIG thermal This command specifies extra memory to be assigned to each zone or gridpoint for a thermal analysis. CONFIG must be given before the BLOCK command. CONFIG

260 3-16 Special Features Structures/Fluid Flow/Thermal/Dynamics thermal can be combined with other calculation modes described in Section 1 in the Command Reference. FLUID HISTORY INITEM PLOT PRINT dtable n Fluid density is a function of temperature. Table number n is used to look up variations of fluid density as a function of temperature. <keyword> temperature thtime x, y value xl xu yl yu history of the temperature at a gridpoint history of real time for heat transfer problems The temperature is set to value at all corners and gridpoints in the range xl x xu, yl y yu. Thermal stresses are not induced by this method of setting the temperature. keyword The following keywords have been added. boundary thermal plots thermal boundaries. hist temperature tfix n1<n2...> keyword <range...> temperature histories n1,n2...assigned by the HIST command temperature contours plots locations of gridpoints with fixed boundaries. gridpoint temp gridpoint temperature hist <n1...> The column headings are: (1) gridpoint address; (2) x-coordinate of gridpoint; (3) y-coordinate of gridpoint; and (4) temperature at gridpoint. Histories n1, n2,...are printed. If no history number is specified, then a list of all history locations is printed.

261 THERMAL ANALYSIS 3-17 prop thermal The properties relevant to the thermal model (and described under the PROPERTY command) are printed. thermal Thermal boundary conditions and sources are printed.. PROPERTY material n keyword v <keyword v> The following keywords have been added. cond thermal conductivity specheat thexp xcond ycond specific heat linear thermal expansion coefficient thermal conductivity in x-direction thermal conductivity in y-direction The actual properties used by the program are the thermal conductivities in the x- and y-directions. The cond keyword simply sets the conductivities in both directions equal to the set value. For jcons = 2 or 5, joint permeability can be specified as a function of temperature by the following keyword. ktable n Table n (see the TABLE command) contains a list of pairs (e.g., permeability and temperature) defining the temperature dependency. The table applies to all joint permeabilities, regardless of joint material number. Note that for a parallel-plate joint, jperm = (1/12) μ, in which μ is the fluid dynamic viscosity. RESET RUN hist All current histories are lost. <<keyword value>...> This command executes thermal timesteps. Calculation is performed until some limiting condition is reached. The limiting condition may be the temperature increase at any point, the number of steps, or the simulated age. The limits are changed by the optional keywords listed below. Once a particular limit is specified, it is used for future RUN commands.

262 3-18 Special Features Structures/Fluid Flow/Thermal/Dynamics age t thermal heating time limit (in consistent units with input properties) delt dt The thermal timestep, dt, is calculated automatically by the program. This parameter allows the user to change the timestep. If the program determines that this value is too large when the explicit scheme is used, it will automatically reduce the timestep to a suitable value when it begins the analysis. The value determined by the program is usually one-half the critical value for numerical stability. If the program selects a value that causes instability, this option can be used to further reduce the timestep. noage step turns off the previously requested test for exceeding age t. The default for the age parameter is that the age is not tested until an age has been explicitly requested via an age = value following a RUN command. s thermal step limit (default = 100,000) temp dtp maximum total temperature change, dtp since the previous mechanical cycles (default is dtp = 20) Two other keywords are available: implicit tol uses the implicit scheme rather than the default explicit scheme. tol Points in this tolerance are merged for thermal calculations (default = 0.1). Old limits apply when set or restarted. When a RUN command has been completed, the program will indicate which parameter has caused it to terminate. To ensure that it stops for the correct one, the values of the others should be set very high. The explicit scheme is always used unless the keyword implicit follows the RUN command. SET keyword value The following keywords have been added.

263 THERMAL ANALYSIS 3-19 nmech nther maximum number of mechanical steps executed between thermal steps, when nther is nonzero (see below). The mechanical stepping will stop when either the maximum step number defined by nmech is reached or the maximum unbalanced force ratio becomes smaller than The default value = 500. number of thermal steps to do before switching to mechanical steps NOTE: The default value of nther is zero, in which case no interlinking occurs. If nther is not zero, the program will switch to mechanical steps every nther steps or when the temperature change parameter (RUN temp = value) is violated. If the temperature change parameter is violated when nther = 0, thermal cycling stops, and further thermal or mechanical cycling is controlled by the user. CAUTION: Geometry changes are ignored by the thermal model until a RUN command is given. This means that when the mechanical models are accessed automatically, the geometry changes are ignored on return to thermal steps. If large geometry changes occur, it is better to divide the run into several RUN commands instead of only one. thdt The thermal timestep is set to value. NOTE: The program calculates the thermal timestep automatically. This keyword allows the user to choose a different timestep. For the explicit method, if the program determines that the chosen step is too large, it will automatically reduce it to a suitable value when thermal steps are taken. It will not revert to a user-selected value until another SET thdt command is issued. The program selects a value that is usually one-half the critical value for numerical stability. This command has the same effect as a RUN delt command. TADD ntab xc,yc ang1,ang2 ntab table number (between 1 and 10) xc,yc coordinates of center of arc ang1,ang2 beginning and ending angles of arc (between 180 and 180 ) Temperatures can be incremented in an angular region using this command. The temperatures are taken from table ntab (see the TABLE command). The angular region is centered at (xc,yc), and the arc is defined by the angles ang1 and ang2. Ifa complete circular region is required, the angles should be given as 180 and 180. The x,y pairs in table ntab represent pairs of radii and temperature increments. The radii represent the distance from (xc,yc), and the code interpolates between these x-values to add to the y-values in the table to the current temperatures. The thermal stresses are also applied, based on these temperature changes.

264 3-20 Special Features Structures/Fluid Flow/Thermal/Dynamics NOTE: The TABLE command must precede the TADD command. TFIX value <range...> The temperatures at all corners and gridpoints are held fixed at value during the simulation. If value is not the current temperature, stresses are induced by the difference between value and the current temperature. An optional range can be given to limit the range of TFIX. (See Section in the Command Reference.) NOTE: By default, all temperatures are free to change initially. TFREE <range...> The temperatures at all corners and gridpoints are allowed to change during the simulation. An optional range can be specified to limit the range of TFREE. (See Section in the Command Reference.) NOTE: By default, all temperatures are free to change initially. THAPP keyword = value1, value2 <range...> The THAPP command applies a thermal boundary condition to external boundaries and thermal sources to internal regions. An optional range can be specified to limit the range of THAPP. (See Section in the Command Reference.) The following keywords are available. convection value1 is the convective heat transfer coefficient (w/m 2 C). value2 is the temperature of the medium to which convection occurs. A convective boundary condition is applied between corners within the range. flux value1 is the initial flux (watts/m 2 ). value2 is the decay constant (s 1 ). A flux boundary condition is applied between corners within the range. If a flux is applied between two blocks, the specified flux will be applied to both blocks. radiation value1 is the radiative heat transfer coefficient. (For black bodies, this is the Stefan-Boltzmann constant, w/m 2 K 4.) value2 is the temperature of the medium to which radiation occurs. A radiation boundary condition is applied between corners within the range.

265 THERMAL ANALYSIS 3-21 source value1 is the initial strength. value2 is the decay constant (s 1). The source keyword results in a volume source of the stated strength in all blocks that have centroids in the specified range. The user is responsible for determining the strength of the source for different size blocks. The initial strength to be given for each block is the intended power/volume ratio multiplied by the area of the block. The correct units for source are Watts/m, (cal/s)/cm or the British equivalents. The decay constant in the source and flux options is defined by the equation Scurr =S ini exp[c d (tcurr -t ini )] where Scurr = current strength; S ini = initial strength; c d = decay constant; tcurr = current time; and t ini = initial time (when THAPP is invoked). To remove a convection or radiation boundary condition, the same condition should be applied with the heat transfer coefficient of opposite sign. CAUTION: It is not physically realistic to use negative heat transfer coefficients in any other circumstances. To remove a flux or source condition, the condition should be applied with the strength replaced by Srep, where Srep =-S ini exp[c d (tcurr -t ini )] Note that unless otherwise specified by the THAPP command, all boundaries are adiabatic (i.e., insulated).

266 3-22 Special Features Structures/Fluid Flow/Thermal/Dynamics FISH Variables The following scalar variables are available in a FISH function to assist with thermal analysis. thdt thtime timestep for the thermal calculation (as set by the SET thdt command) thermal time The UDEC grid variable, temperature, can be accessed and modified by using the FISH function fmem to access the gridpoint temperature. The temperature is found from fmem(gp + $KGTEMP) where gp is the gridpoint index, and $KGTEMP is the symbolic name for the gridpoint temperature address. See Section 4 in the FISH volume and the data files in Section 3.6. Also, thermal property values may be accessed (changed, as well as tested) in a FISH function. See the PROPERTY command in Section for a list of the thermal properties.

267 THERMAL ANALYSIS Systems of Units for Thermal Analysis All thermal quantities must be given in an equivalent set of units. No conversions are performed by the program. Tables 3.2 and 3.3 present examples of consistent sets of units for thermal parameters. Table 3.2 System of SI units for thermal problems Length m m m cm Density kg/m kg/m kg/m g/cm 3 Stress Pa kpa MPa bar Temperature K K K K Time s s s s Specific Heat J/(kg K) 10 3 J/(kg K) 10 6 J/(kg K) 10 6 cal/(g K) Thermal Conductivity W/(mK) W/(mK) W/(mK) (cal/s)/cm 2 K 4 Convective Heat Transfer W/(m 2 K) (W/m 2 K) W/(m 2 K) (cal/s)/(cm 2 K) Coefficient Radiative Heat Transfer W/(m 2 K 4 ) W/(m 2 K 4 ) W/(m 2 K 4 ) (cal/s)/cm 2 K 4 Coefficient Flux Strength W/m 2 W/m 2 W/m 2 (cal/s)/cm 2 Source Strength W/m 3 W/m 3 W/m 3 (cal/s)/cm 3 Decay Constant s 1 s 1 s 1 s 1 Stefan-Boltzmann Constant W/m 2 K 4 W/m 2 K 4 W/m 2 K 4 cal/(cm 2 sk 4 ) Table 3.3 System of Imperial units for thermal problems Length ft in Density slugs/ft 3 snails/in 3 Stress lbf psi Temperature R R Time hr hr Specific Heat (32.17) 1 Btu/(1b R) (32.17) 1 Btu/(1b R) Thermal Conductivity (Btu/hr)/(in R) (Btu/hr)/(in R) Convective Heat Transfer Coefficient (Btu/hr)/(ft 2 R) (Btu/hr)/(in 2 R) Radiative Heat Transfer Coefficient (Btu/hr)/(ft 2 R 4 ) (Btu/hr)/(in 2 R 4 ) Flux Strength (Btu/hr)/ft 2 (Btu/hr)/in 2 Source Strength (Btu/hr)/ft 3 (Btu/hr)/in 3 Decay Constant hr 1 hr 1 Stefan-Boltzmann Constant Btu/(ft 2 hr R 4 ) Btu/(in 2 hr R 4 )

268 3-24 Special Features Structures/Fluid Flow/Thermal/Dynamics where 1K = 1.8 R; 1J = cal = Btu; 1J/kg K = btu/1b R; 1W = 1 J/s = cal/s = Btu/hr; 1W/m K = Btu/(ft/hr R); and 1W/m 2 K = Btu/ft 2 hr R. Note that, unless radiation is being used, temperatures may be quoted in the more common units of C (instead of K)or F (instead of R), where Temp( C) = 5 9 (Temp( F ) 32); Temp( F ) = (1.8 Temp( C)) + 32; Temp( C) = Temp(K) 273; and Temp( F ) = Temp(R) 460.

269 THERMAL ANALYSIS Verification Examples Several verification examples are presented to demonstrate the thermal model in UDEC. The data files for these examples are located in the Datafiles\Thermal directory. All of the models contain joints. This allows the evaluation of joint behavior on the thermal and thermal-mechanical response of the models. The joint stiffnesses can influence the results for thermal-mechanical analyses; stiffnesses are at least two to three orders of magnitude higher than the block stiffnesses for the thermal-mechanical examples Conduction through a Composite Wall An infinite wall consisting of two distinct layers is exposed to an atmosphere at a high temperature on one side and a low temperature on the other. The wall eventually reaches an equilibrium at a constant heat flux and unchanging temperature distribution. The two layers of the wall have the specifications presented in Table 3.4. Figure 3.6 shows the wall geometry and boundary conditions. The wall is of infinite height and thickness, and the temperatures of the atmosphere on either side are constant. The two layers, individually, are homogeneous and isotropic, and the conductive contact between them is perfect. Table 3.4 Problem specifications temperature of outside T i = 3000 C T o =25 C convection coefficient h i = 100 w/m 2 C h o =15w/m 2 C thermal conductivity k 1 = 1.6 w/m C k 2 = 0.2 w/m C thickness d 1 =25cm d 2 =15cm k = k = T 1 T 2 T 3 h = 15 h i = 100 o T i = 3000 C d = 25 cm 1 2 d = 15 cm o T = 25 C o o Figure 3.6 Composite wall

270 3-26 Special Features Structures/Fluid Flow/Thermal/Dynamics The analytical steady-state solution to this problem is quite simple and common. The total equilibrium heat flux is where R T is the sum of the four thermal resistances: R 1 =1/h i R 2 = d 1 /k 1 R 3 = d 2 /k 2 R 4 =1/h o q A = T i T o R T (3.21) This heat flux is constant across the three interfaces. Hence, after setting this flux equal to the temperature difference divided by the interface resistance and solving for the unknown, we arrive at T 1 = T i q A 1 h i T 2 = T 1 q A d1 k 1 (3.22) T 3 = T 2 q A d2 k 2 The temperature will vary linearly among the three. Example 3.1 contains the input commands necessary to solve this problem with UDEC. Commands for both the explicit and implicit solutions are given. The data file as shown runs the explicit solution. If the semicolon is removed from the SET thdt=6000 command, and the implicit keyword is added to the RUN command, then the implicit solution will be performed. The model is run to a thermal age of 600,000 seconds to reach steady-state for both solutions.

271 THERMAL ANALYSIS 3-27 Example 3.1 Conduction through a composite wall config thermal round edge block 0,0 0,2.5E-2 0.4,2.5E-2 0.4,0 crack (0.25,-1) (0.25,1) join gen edge 0.05 range 0,0.25 0,2.5E-2 gen edge range 0.25,0.4 0,2.5E-2 group zone wall:high density range 0,0.25 0,2.5E-2 group zone wall:low density range 0.25,0.4 0,2.5E-2 zone model elastic density 1E4 cond 1.6 specheat 300 range group & wall:high density zone model elastic density 1.25E3 cond 0.2 specheat 300 range group & wall:low density thapp convection range , ,2.6E-2 thapp convection range 0.399, ,2.6E-2 initemp range , ,2.6E-2 initemp range 0.249, ,2.51E-2 run age temp tol 1.0E-4 save cond1.sav ; def constants d_1 = 0.25 d_2 = 0.15 t_i = t_o = 25.0 h_i = h_o = 15.0 k_1 = 1.6 k_2 = 0.2 r_1 = 1.0/h_i r_2 = d_1/k_1 r_3 = d_2/k_2 r_4 = 1.0/h_o r_t = r_1 + r_2 + r_3 + r_4 q_a = (t_i-t_o) / r_t t_1 = t_i - q_a * r_1 t_2 = t_1 - q_a * r_2 t_3 = t_2 - q_a * r_3 end constants ; store numerical results in table 1 call block.fin def num_sol

272 3-28 Special Features Structures/Fluid Flow/Thermal/Dynamics ib = block_head loop while ib # 0 ig = b_gp(ib) loop while ig # 0 x = gp_x(ig) tmp = fmem(ig+$kgtemp) table(1,x) = tmp ig = gp_next(ig) endloop ib = b_next(ib) endloop end ; ; store analytical results in table 2 def ana_sol ib = block_head loop while ib # 0 ig = b_gp(ib) loop while ig # 0 x = gp_x(ig) if x < d_1 then tmp = x * ((t_2-t_1)/d_1) + t_1 else x_2 = x - d_1 tmp = x_2 * ((t_3-t_2)/d_2) + t_2 endif table(2,x) = tmp ig = gp_next(ig) endloop ib = b_next(ib) endloop end num_sol ana_sol save cond2.sav ret The wall is idealized by the geometry shown in Figure 3.7. Since the model is infinitely long in one direction, the model is essentially one-dimensional, and horizontal boundaries may be represented as adiabatic boundaries.

273 THERMAL ANALYSIS 3-29 Adiabatic Boundary Interface Convective Boundary Figure 3.7 Idealization of the wall for the UDEC model In the UDEC analysis, the wall is defined by two deformable blocks; each block corresponds to an individual layer of the wall. The zoning for the two blocks is shown in Figure 3.8. An adiabatic boundary condition (zero heat flux across boundary) is applied to the top and bottom of this model to simulate the infinite dimensions of the wall. (Adiabatic boundaries are the default condition.) The appropriate convective boundary conditions are applied to the ends of the grid, and the different sets of thermal properties are applied to the two blocks to model the composite material. JOB TITLE : Conduction through a Composite Wall - explicit solution UDEC (Version 5.00) LEGEND Oct :00:00 cycle 0 time 0.000E+00 sec thermal time = 6.000E+05 sec zones in fdef blocks block plot Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 3.8 Zone distribution Figure 3.9 shows a contour plot of the steady-state temperature distribution using the explicit solution.

274 3-30 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE : Conduction through a Composite Wall - explicit solution UDEC (Version 5.00) LEGEND Oct :01:00 cycle 0 time 0.000E+00 sec thermal time = 6.000E+05 sec grid point temperature cont contour interval= 5.000E E+02 to 2.500E E E E E E+03 block plot Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 3.9 Steady-state temperature distribution Figure 3.10 compares UDEC s temperature distribution with the analytical solution. The numerical calculations for steady-state temperatures are stored in table 1, and the analytical values are stored in table 2 for comparison. UDEC (Version 5.00) LEGEND 13-Oct :02:00 cycle 0 time 0.000E+00 sec thermal time = 6.000E+05 sec table plot Temperature - UDEC Temperature - Analytic X X X

275 THERMAL ANALYSIS 3-31 Table 3.5 displays a more precise comparison for five points along the wall, including the three interface points ( interface refers to thermal properties, not a mechanical interface). The results based on the implicit solution are identical. Table 3.5 Comparison of UDEC results and the analytical solution Position Analytical ( C) UDEC ( C) % Error T < 0.01% < 0.01% T T < 0.01% The comparison between UDEC and the analytical solution shows that, for this simple onedimensional problem, UDEC produces excellent agreement. The errors on both the boundaries and the interface are negligible (<0.1%).

276 3-32 Special Features Structures/Fluid Flow/Thermal/Dynamics Thermal Response of a Heat-Generating Slab An infinite plate of thickness 2L = 1 m generates heat internally. This problem determines the transient temperature distribution after application of a constant temperature boundary condition. The physical properties of the plate in question are density (ρ) 500 kg/m 3 specific heat (C p ) thermal conductivity (k) 0.2 J/kg C 20 w/m C The plate is initially at a uniform temperature of 60 C, the surface is then fixed at 32 C, and the plate itself has internal heat generation of 40 kw/m 3 (as shown in Figure 3.11). q = 40 kw/m 3 32 o C 32 o C 1 m Figure 3.11 Heat-generating slab showing initial and boundary conditions Assuming that the slab is infinitely long and that the material is homogeneous, isotropic and continuous, with temperature-independent thermal properties, the governing equation for this problem is 2 T x 2 + Q k = 1 κ T t (3.23) where T = temperature; x = distance from slab centerline; Q = constant volumetric heat generation rate; k = thermal conductivity; t = time; and κ = diffusivity = k ρ C p.

277 THERMAL ANALYSIS 3-33 By symmetry, only half of the plate is modeled. The applied initial and boundary conditions are T x = 0 x = 0; t>0 T = T s x = L; t>0 T = T i t 0 where T i = initial uniform temperature; T s = constant temperature at the slab faces; and L = slab half-width. The integration of Eq. (3.23) is presented by Ozisik (1980): T(x,t) = T s + Q 2k (L2 x 2 ) + 2 L (T i T s ) ( ) ( 1) m e κβ2 m t cos(βm x) m=0 β m (3.24) 2Q Lk ( ) ( 1) m e κβ2 m t cos(βm x) m=0 where β m are the positive roots of the transcendental equation β 3 m cos(β m L) = 0 or β m = (2m + 1)π 2L,m= 0, 1, 2... (3.25) For steady-state condition (t ), the two last terms of Eq. (3.24) tend to zero, so T steady (x) = T s + Q 2k (L2 x 2 ) (3.26) The conditions imposed to model this problem are shown in Figure 3.12; the corresponding UDEC model is given in Figure Because the plate is infinitely long, and because the heat generation is uniform, symmetry conditions exist for any plane perpendicular to the long axis of the plate. These are represented by adiabatic boundaries. The right boundary of the model has a fixed temperature of 32 C. The left boundary is a symmetry line.

278 3-34 Special Features Structures/Fluid Flow/Thermal/Dynamics adiabatic boundary q = 40 kw/m 3 fixed temp. 32-C 0.5m Figure 3.12 Model conditions for heat-generating slab JOB TITLE : Heat-Generating Slab UDEC (Version 5.00) LEGEND Oct :43:53 cycle 0 time 0.000E+00 sec thermal time = 0.000E+00 sec zones in fdef blocks block plot Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 3.13 UDEC zone distribution The model is run to a thermal time of 0.1, 0.5 and 5 seconds. The last time corresponds to a steady-state condition. Example 3.2 contains the input commands necessary to solve this problem with UDEC.

279 THERMAL ANALYSIS 3-35 Example 3.2 Thermal response of a heat-generating slab title Thermal Response of a Heat-Generating Slab - Explicit Solution ; --- problem constants --- def constants tabo = -1 tabe = 0 v_l = 0.5 v_ti = 60. v_ts = 32. v_q = 4.e4 v_k = 20. v_cp = 0.2 v_rho = 500. v_kappa = v_k / (v_rho * v_cp) qk = v_q / v_k tits = v_ti - v_ts pi2l = pi / (2.0 * v_l) end constants ; store analytical results in odd numbered tables ; store numerical results in even numbered tables call block.fin def solution tabo = tabo + 2 tabe = tabe + 2 ib = block_head loop while ib # 0 ig = b_gp(ib) loop while ig # 0 if gp_y(ig) > ym_tol then if gp_y(ig) < yp_tol then x_p = gp_x(ig) kt = v_kappa * thtime s_old = 0.0 monem = -1. m = 0 loop while m < 100 monem = -monem betam = pi2l * (2.0 * m + 1.) betam2 = betam * betam e1 = exp(-kt*betam2)/betam s_new = s_old+(tits-qk/betam2)*monem*cos(betam*x_p)*e1 if s_new = s_old then

280

281 THERMAL ANALYSIS 3-37 label table 1 Temp. 0.1 Sec. - UDEC label table 2 Temp. 0.1 Sec. - Analytic label table 3 Temp. 0.5 Sec. - UDEC label table 4 Temp. 0.5 Sec. - Analytic label table 5 Temp. 5.0 Sec. - UDEC label table 6 Temp. 5.0 Sec. - Analytic plot tab 1 cross 2 3 cross 4 5 cross 6 hold ret Figure 3.14 shows the evolution of temperature in the center of the slab (x = 0). Steady-state conditions are reached at t = 5. Figure 3.15 shows the temperature distribution at steady state. The temperature distributions for t = 0.1, 0.5 and 5 seconds are compared to the analytical solution in Figure The agreement is excellent, with an error of less than 1%. JOB TITLE : Heat-Generating Slab UDEC (Version 5.00) (e+002) 3.20 LEGEND 14-Oct :44:43 cycle 0 time 0.000E+00 sec thermal time = 5.000E+00 sec history plot Y-axis: 2 - temperature X-axis: 3 - thermal time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 3.14 Temperature evolution in the center of the slab

282 3-38 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE : Heat-Generating Slab UDEC (Version 5.00) LEGEND Oct :45:08 cycle 0 time 0.000E+00 sec thermal time = 5.000E+00 sec grid point temperature cont contour interval= 4.000E E+01 to 2.800E+02 block plot 4.000E E E E E E E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 3.15 Temperature distribution at steady-state JOB TITLE : Heat-Generating Slab UDEC (Version 5.00) (e+002) 3.20 LEGEND 14-Oct :44:08 cycle 0 time 0.000E+00 sec thermal time = 5.000E+00 sec table plot Temp. 0.1 Sec. - UDEC X X X Temp. 0.1 Sec. - Analytic Temp. 0.5 Sec. - UDEC X X X Temp. 0.5 Sec. - Analytic Temp. 5.0 Sec. - UDEC X X X Temp. 5.0 Sec. - Analytic Vs. 0.00E+00<X value> 5.00E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-001) Figure 3.16 UDEC and analytical temperature distributions at thermal time = 0.1, 0.5 and 5.0 seconds (analytical values = odd-numbered tables; numerical values = even-numbered tables)

283 THERMAL ANALYSIS Heating of a Hollow Cylinder A hollow cylinder of infinite length is initially at a constant temperature of 0 C. The inner radius of the cylinder is exposed to a constant temperature of 100 C, and the outer radius is kept at 0 C. The problem is to determine the temperatures and thermally induced stresses in the cylinder when the equilibrium thermal state is reached. Nowacki (1962) provides the solution to this problem in terms of the temperatures and radial, tangential and axial stresses at the steady-state thermal state: T(r) T a σ r (r) mgt a σ t (r) mgt a σ a (r) mgt a = ln(b/r) ln(b/a) [ ] ln(b/r) = ln(b/a) (b/r)2 1 (b/a) 2 1 [ ln(b/r) 1 = ln(b/a) = [ 2ln(b/r) λ 2(λ+G) ln(b/a) + (b/r)2 + 1 (b/a) 2 1 ] (3.27) (3.28) (3.29) ( )( ) ] λ 2 + 2λ + G (b/a) 2 (3.30) 1 where T = temperature; r = radial distance from the cylinder center; a = inner radius of the cylinder; b = outer radius of the cylinder; T a = temperature at the inner radius; σ r = radial stress; σ t = tangential stress; σ a = axial stress; m = λ+2g 3Kα ; λ = K 2 3 G; K is the bulk modulus; G is the shear modulus; and α is the linear thermal expansion coefficient. The analytical solutions for temperature and stresses are programmed as FISH functions in the UDEC data file. The analytical and numerical results can then be compared directly in tables.

284 3-40 Special Features Structures/Fluid Flow/Thermal/Dynamics The following properties are prescribed for this example. Geometry inner radius of cylinder (a) outer radius of cylinder (b) 1.0 m 2.0 m Material Properties density (ρ) 2000 kg/m 3 specific heat (C p ) J/kg C thermal conductivity (k) 4.2 W/m C linear thermal expansion coefficient (κ) / C shear modulus (G) 28.0 GPa bulk modulus (K) 48.0 GPa A quarter-section of the cylinder is modeled with UDEC. Figure 3.17 shows the UDEC zoning. A constant-temperature boundary of 100 C is specified for the inner radius of the model; the temperature at the outer radius is specified to be 0 C. The UDEC model can be run as either a coupled or uncoupled thermal-mechanical analysis. In this example, we run the model in an uncoupled mode: the thermal calculation is performed first to reach the equilibrium heat flux state; then the thermally induced mechanical stresses are calculated. The UDEC data file is listed in Example 3.3. JOB TITLE : Heating a Hollow Cylinder UDEC (Version 5.00) LEGEND Oct :15:49 cycle 0 time 0.000E+00 sec thermal time = 3.445E+05 sec zones in fdef blocks block plot Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 3.17 UDEC grid for heating of a hollow cylinder

285 THERMAL ANALYSIS 3-41 Example 3.3 Heating of a hollow cylinder config thermal title Heating of a Hollow Cylinder round=.001 ; set geometry (one quarter of a rod) bl arc (0,0) (1.0,0) arc (0,0) (1.25,0) arc (0,0) (1.5,0) arc (0,0) (1.75,0) arc (0,0) (2.0,0) ; hollow out the rod to make a cylinder del range annulus (0,0) ; delete outside of cylinder del range annulus (0,0) ; add construction joints for zoning jset angle 22.5 spacing 4 origin 0,0 jset angle 45 spacing 4 origin 0,0 jset angle 67.5 spacing 4 origin 0,0 join_cont gen quad 0.1 range ann (0,0) gen quad 0.15 range ann (0,0) gen quad 0.2 ; set material properties group zone block zone model elastic density 2E3 bulk 4.8E10 shear 2.8E10 cond 4.2 specheat & 880 thexp 5.4E-6 range group block ; set boundary conditions bound yvel 0.0 range (0,3) (-.01,.01) bound xvel 0.0 range (-.01,.01) (0,3) ; temperature=100 fixed at radius= 1 tfix range ann (0,0) ; temperature=0 fixed at radius= 2 tfix 0.0 range ann (0,0) ; thermal histories to check thermal equilibrium history temperature 1.25,1.25 history temperature 1.0,1.0 history thtime ; mechanical histories at different joints to check mech. equil. hist nstr (1.25,0) hist nstr (1.25,1.25) hist nstr (0,1.25) hist sstr (1.25,0)

286 3-42 Special Features Structures/Fluid Flow/Thermal/Dynamics hist sstr (1.25,1.25) hist sstr (0,1.25) ; run thermal problem until equilibrium (explicit procedure) run temp=15000 step=5000 tol.001 save cy1.sav ; ; then run mechanical problem solve ratio 1e-6 save cy2.sav ; ; --- fish constants --- def constants c_b = 2. eps = 1.e-4 xtol = 0.01 yp_tol = 0.05 c_g = 28e9 ; shear modulus c_k = 48e9 ; bulk modulus c_al = 5.4e-6 ; coefficient of thermal expansion t1 = 100. ; boundary temperature oc1 = 1. / ln(c_b) oc2 = 1. / (c_b * c_b - 1.) oc3 = 0.5 * (c_k - c_g * 2. / 3.)/(c_k + c_g / 3.) c_mmu = c_g * (3. * c_k * c_al) / (c_k + 4. * c_g / 3.) tab1 = 1 ; numerical temperature tab2 = 2 ; analytical temperature tab3 = 3 ; numerical radial stress tab4 = 4 ; analytical radial stress tab5 = 5 ; numerical tangential stress tab6 = 6 ; analytical tangential stress tab7 = 7 ; numerical axial stress tab8 = 8 ; analytical axial stress end constants call block.fin def num_solt ib = block_head loop while ib # 0 ig = b_gp(ib) loop while ig # 0 if gp_y(ig) < eps then x = gp_x(ig) table(tab1,x) = fmem(ig+$kgtemp) / t1 endif ig = gp_next(ig) endloop

287 THERMAL ANALYSIS 3-43 ib = b_next(ib) endloop end def ana_solt nn = 0 ib = block_head loop while ib # 0 ig = b_gp(ib) loop while ig # 0 if gp_y(ig) < eps then x = gp_x(ig) table(tab2,x) = ln(c_b / x) * oc1 nn = nn + 1 endif ig = gp_next(ig) endloop ib = b_next(ib) endloop end ; def num_solst ; table tab1 must be available ns = 1 nz = 0 loop while ns < nn x = (xtable(tab1,ns) + xtable(tab1,ns+1)) * 0.5 xp_tol = x + xtol xm_tol = x - xtol ib = block_head loop while ib # 0 iz = b_zone(ib) loop while iz # 0 if z_x(iz) < xp_tol then if z_x(iz) > xm_tol then if z_y(iz) < yp_tol then nz = nz + 1 xc = z_x(iz) yc = z_y(iz) ra2 = xc*xc + yc*yc ra = sqrt(ra2) xtable(tab3,nz) = ra xstr = z_sxx(iz)*xc*xc ystr = z_syy(iz)*yc*yc xystr = 2.*z_sxy(iz)*xc*yc val = (xstr + ystr + xystr)/ra2 ytable(tab3,nz) = val / (c_mmu * t1) xtable(tab5,nz) = ra

288 3-44 Special Features Structures/Fluid Flow/Thermal/Dynamics xstr = z_sxx(iz)*yc*yc ystr = z_syy(iz)*xc*xc xystr = 2.*z_sxy(iz)*xc*yc val = (xstr + ystr - xystr)/ra2 ytable(tab5,nz) = val / (c_mmu * t1) xtable(tab7,nz) = ra val = z_szz(iz) ytable(tab7,nz) = val / (c_mmu * t1) endif endif endif iz = z_next(iz) end_loop ib = b_next(ib) end_loop ns = ns + 1 end_loop end def ana_solst ; table tab1 must be available ns = 1 nz = 0 loop while ns < nn x = (xtable(tab1,ns) + xtable(tab1,ns+1)) * 0.5 xp_tol = x + xtol xm_tol = x - xtol ib = block_head loop while ib # 0 iz = b_zone(ib) loop while iz # 0 if z_x(iz) < xp_tol then if z_x(iz) > xm_tol then if z_y(iz) < yp_tol then nz = nz + 1 xc = z_x(iz) yc = z_y(iz) ra = sqrt(xc*xc + yc*yc) xtable(tab4,nz) = ra val = c_b / ra ytable(tab4,nz) = -(ln(val)*oc1-(val*val-1.)*oc2) xtable(tab6,nz) = ra ytable(tab6,nz) = -((ln(val)-1.)*oc1+(val*val+1.)*oc2) xtable(tab8,nz) = ra ytable(tab8,nz) = -((2.*ln(val)-oc3)*oc1+2.*oc3*oc2) endif endif endif

289 THERMAL ANALYSIS 3-45 iz = z_next(iz) end_loop ib = b_next(ib) end_loop ns = ns + 1 end_loop end save cy3.sav ; num_solt ana_solt num_solst ana_solst label table 1 Temperature - UDEC label table 2 Temperature - Analytic label table 3 Radial Stress - UDEC label table 4 Radial Stress - Analytic label table 5 Tangential Stress - UDEC label table 6 Tangential Stress - Anal label table 7 Axial Stress - UDEC label table 8 Axial Stress - Analytic pl tab 1 cross 2 pl tab 3 cross 4 pl tab 5 cross 6 pl tab 7 cross 8 save cy4.sav Numerical and analytical results are compared in Figures 3.18 through The figures show plots of tables for temperature and stress distributions through the cylinder at steady state. In each figure, the numerical values are plotted as the odd numbered table, and the analytical values are plotted as the even numbered table. The plotted values are normalized. Temperature is normalized by dividing by T a, and stress is normalized by dividing by mgt a. Figure 3.18 shows the temperature distribution at steady state for the numerical and analytical solutions. Comparisons of results for radial, tangential and axial stress distributions at steady state are provided in Figures 3.19, 3.20 and 3.21, respectively.

290 3-46 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE : Heating a Hollow Cylinder UDEC (Version 5.00) 1.20 LEGEND 14-Oct :02:54 cycle time 1.587E-01 sec thermal time = 3.445E+05 sec table plot Temperature - UDEC X X X Temperature - Analytic Vs. 1.00E+00<X value> 2.00E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 3.18 Temperature distribution at steady state for heating of a hollow cylinder JOB TITLE : Heating a Hollow Cylinder UDEC (Version 5.00) (e-001) LEGEND 14-Oct :02:54 cycle time 1.587E-01 sec thermal time = 3.445E+05 sec table plot Radial Stress - UDEC X X X Radial Stress - Analytic Vs. 1.04E+00<X value> 1.94E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 3.19 Radial stress distribution at steady state for heating of a hollow cylinder

291 THERMAL ANALYSIS 3-47 JOB TITLE : Heating a Hollow Cylinder UDEC (Version 5.00) 0.80 LEGEND 14-Oct :02:54 cycle time 1.587E-01 sec thermal time = 3.445E+05 sec table plot Tangential Stress - UDEC X X X Tangent. Stress - Analytic Vs. 1.04E+00<X value> 1.94E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 3.20 Tangential stress distribution at steady state for heating of a hollow cylinder JOB TITLE : Heating a Hollow Cylinder UDEC (Version 5.00) 0.20 LEGEND 14-Oct :02:54 cycle time 1.587E-01 sec thermal time = 3.445E+05 sec table plot Axial Stress - UDEC X X X Axial Stress - Analytic Vs. 1.04E+00<X value> 1.94E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 3.21 Axial stress distribution at steady state for heating of a hollow cylinder

292 3-48 Special Features Structures/Fluid Flow/Thermal/Dynamics Infinite Line Heat Source in an Infinite Medium An infinite line heat source with a constant heat-generating rate is located in an infinite elastic medium with constant thermal properties. Nowacki (1962) provides the solution to this problem for the transient values of temperature, radial and tangential stress and radial displacement: T a = 1 4π E 1 (ξ) (3.31) σ r bg = 1 4π σ t bg = 1 4π u r bl = 1 8π r [E 1 (ξ) + 1 ] e ξ ξ [E 1 (ξ) 1 ] e ξ ξ [E 1 (ξ) + 1 ] e ξ ξ (3.32) (3.33) (3.34) where ξ = 4κt r2 r = radial distance to the line source; κ = ρc k p ; a = q k ; b = αa3k+4g 9K ; L = unit length; and E 1 (ξ) = ξ e u u du is the exponential integral. The material properties and initial and boundary conditions for this example are defined as follows. Material Properties density (ρ) 2000 kg/m 3 shear modulus (G) 30 GPa bulk modulus (K) 50 GPa specific heat (C p ) 1000 J/kg C thermal conductivity (k) 4 W/m C linear thermal expansion coefficient (α) / C

293 THERMAL ANALYSIS 3-49 Initial/Boundary Conditions initial uniform temperature initial stress state Line Heat Source energy release per unit length (Q) 0 C no stresses 1600 W/m It is assumed that the material properties are temperature-independent, the thermal output of the source is constant (no decay), and the heat line source is of infinite length. The UDEC model for this problem is a quarter-section of a cylindrical disk with a hole in the center. The axis of the line heat source coincides with the centroid of the disk. The zoning in the model is radially graded in the xy-plane by cutting the original block with a series of construction arcs; each arc has a radius that is 1.1 times larger than the previous arc. The model is shown in Figure A close-up view of the block zoning near the heat source is shown in Figure JOB TITLE : Line Heat Source (*10^2) UDEC (Version 5.00) LEGEND 14-Oct :56:21 cycle 0 time 0.000E+00 sec thermal time = 0.000E+00 sec zones in fdef blocks block plot Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^2) Figure 3.22 UDEC grid for an infinite line heat source

294 3-50 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE : Line Heat Source (*10^1) UDEC (Version 5.00) LEGEND 14-Oct :35:44 cycle time 4.023E+00 sec thermal time = 3.110E+07 sec zones in fdef blocks block plot Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 3.23 Close-up view of zoning in blocks The line heat source is simulated by a constant heat flux applied at the inner hole boundary of the disk. The line heat source is assumed to have a fictitious radius of R = 1.0 m, so that the applied flux will be Flux = q = Q 2πR = w/m2 The other boundaries of the model are kept adiabatic to represent thermal symmetry planes. The disk is extended to a radius of 500 m to simulate infinity. The far boundary is mechanically fixed; the boundaries along the x-axis and y-axis are fixed to represent shear-free symmetry planes. The problem is first solved thermally to an age of one year using the implicit solution algorithm, and then stepped to mechanical equilibrium. The UDEC data file is listed in Example 3.4. The dimensionless form of the analytical solutions in Eqs. (3.31) to (3.34) are programmed as FISH functions in Example 3.4. The analytical and numerical values can then be compared directly in tables. The analytical solutions for temperature and radial displacement are programmed in FISH function ana soltu, and for radial and tangential stresses in ana solst. The exponential integral function used in the analytical solutions is programmed as a separate FISH function contained in file EXP INT.FIS (see Example 3.5). The dimensionless values for the numerical results for temperature and displacement are calculated in FISH function num soltu, and for radial and tangential stresses in num solst. The numerical values for dimensionless temperature, radial stress,

295 THERMAL ANALYSIS 3-51 tangential stress and radial displacement are stored in Tables 1, 3, 5 and 7, respectively. The analytical values for dimensionless temperature, radial stress, tangential stress and radial displacement are stored in Tables 2, 4, 6 and 8, respectively. Example 3.4 Infinite line heat source in an infinite medium new ;file: linesource.dat title Infinite line heat source in infinite elastic medium config thermal round 1E-3 edge 2E-3 block 0,0 0, , ,0 ;Name:arc_cut ;Input:xcut/float/1.0/x-coord of cut ;Input:narc/int/6/number of segments in arc ;Input:ntot/int/48/number of arcs ;Input:rat/float/1.1/geometric ratio def arc_cut nc = 1 xloc = xcut numarc = narc loop while nc < ntot command arc (0,0) (xloc,0) 90 numarc endcommand xcut = rat * xcut xloc = xloc + xcut nc = nc + 1 endloop end set xcut=1.0 narc=6 ntot=48 rat=1.1 arc_cut delete range annulus (0,0) delete range annulus (0,0) 0 1 jset angle 45 spacing 800 origin 0,0 join_cont gen quad ; ; set material properties group zone block zone model elastic density 2E3 bulk 5E10 shear 3E10 cond 4 specheat 1E3 & thexp 5E-6 range group block ; ; set boundary conditions

296 3-52 Special Features Structures/Fluid Flow/Thermal/Dynamics boundary yvelocity 0 range 0, ,0.1 boundary xvelocity 0 range -0.1,0.1 0,500 boundary nvelocity 0 range annulus (0,0) ; apply heat source for 1600 W/m thapp flux ,0 range annulus (0,0) ; ; thermal histories to check thermal equilibrium history temperature 1.25,1.25 history temperature 5.0,5.0 history thtime ; ; mechanical histories at different joints to check mech. equil. hist nstr (1.25,0) hist nstr (1.25,1.25) hist nstr (0,1.25) hist sstr (1.25,0) hist sstr (1.25,1.25) hist sstr (0,1.25) ; save line.sav ; ; run thermal problem for 1 year of heating (implicit procedure) set thdt= run implicit step 4800 temp save line_th.sa ; res line_th.sav ; ; then run mechanical problem set ovtol 0.1 frac damp auto step 9000 ; save line_1yr.sav ; ; ; line source in infinite medium ; comparison of numerical and analytical solutions ; ; --- fish functions --- def cons ; t_time = 3.11e7 xtol = 0.3 yp_tol = 5.0 eps = 1.e-4

297 THERMAL ANALYSIS 3-53 c_g = 3.e10 ; shear modulus c_k = 5.e10 ; bulk modulus c_al = 5.e-6 ; coefficient of thermal expansion c_tk = 4. ; conductivity c_cp = 1e3 ; specific heat q_q = ; line source intensity c_density = 2.e3 kappa = c_tk / (c_density * c_cp) o4c = 1. / (4. * kappa) o4p = 1. / (4. * pi) o8p = o4p * 0.5 val = c_k / c_g c_nu = (3.*val-2.)/(6.*val+2.) c_eta = c_al * c_g * (1.+c_nu)/(1.-c_nu) a_a = q_q / c_tk b_b = c_eta * a_a c_c = b_b * 1.0 / c_g a_a = 1. / a_a b_b = 1. / b_b c_c = 1. / c_c tab1 = 1 ; numerical temperature tab2 = 2 ; analytical temperature tab3 = 3 ; numerical radial stress tab4 = 4 ; analytical radial stress tab5 = 5 ; numerical tangential stress tab6 = 6 ; analytical tangential stress tab7 = 7 ; numerical radial displacement tab8 = 8 ; analytical radial displacement end cons call block.fin call exp_int.fis def num_soltu nn = 0 ib = block_head loop while ib # 0 ig = b_gp(ib) loop while ig # 0 if gp_y(ig) < eps then x = gp_x(ig) if x > 0.0 then table(tab1,x) = fmem(ig+$kgtemp) * a_a table(tab7,x) = gp_xdis(ig) * c_c nn = nn + 1 end_if end_if

298 3-54 Special Features Structures/Fluid Flow/Thermal/Dynamics ig = gp_next(ig) endloop ib = b_next(ib) end_loop end num_soltu def ana_soltu ib = block_head loop while ib # 0 ig = b_gp(ib) loop while ig # 0 if gp_y(ig) < eps then x = gp_x(ig) if x > 0.0 then e_val = x * x * o4c / thtime val = exp_int table(tab2,x) = val * o4p table(tab8,x) = (val + (1.-exp(-e_val))/e_val) * x * o8p end_if endif ig = gp_next(ig) endloop ib = b_next(ib) end_loop end ; def num_solst ; table tab1 must be available ns = 2 nz = 0 loop while ns < nn x = (xtable(tab1,ns) + xtable(tab1,ns+1)) * 0.5 if ns > 10 then xtol = 1.0 endif if ns > 50 then xtol = 5.0 endif xp_tol = x + xtol xm_tol = x - xtol ib = block_head loop while ib # 0 iz = b_zone(ib) loop while iz # 0 if z_x(iz) < xp_tol then if z_x(iz) > xm_tol then nzgp = 0

299 THERMAL ANALYSIS 3-55 igz1 = z_gp(iz,1) ygz1 = gp_y(igz1) if ygz1 < eps then nzgp = nzgp + 1 endif igz2 = z_gp(iz,2) ygz2 = gp_y(igz2) if ygz2 < eps then nzgp = nzgp + 1 endif igz3 = z_gp(iz,3) ygz3 = gp_y(igz3) if ygz3 < eps then nzgp = nzgp + 1 endif if nzgp = 2 then nz = nz + 1 xc = z_x(iz) yc = z_y(iz) ra2 = xc*xc + yc*yc ra = sqrt(ra2) xstr = z_sxx(iz)*xc*xc ystr = z_syy(iz)*yc*yc xystr = 2.*z_sxy(iz)*xc*yc val = (xstr + ystr + xystr)/ra2 table(tab3,ra) = val * b_b xstr = z_sxx(iz)*yc*yc ystr = z_syy(iz)*xc*xc xystr = 2.*z_sxy(iz)*xc*yc val = (xstr + ystr + xystr)/ra2 table(tab5,ra) = val * b_b endif endif endif iz = z_next(iz) end_loop ib = b_next(ib) end_loop ns = ns + 1 end_loop end def ana_solst ; table tab1 must be available ns = 2 nz = 0 loop while ns < nn x = (xtable(tab1,ns) + xtable(tab1,ns+1)) * 0.5

300 3-56 Special Features Structures/Fluid Flow/Thermal/Dynamics if ns > 10 then xtol = 1.0 endif if ns > 50 then xtol = 5.0 endif xp_tol = x + xtol xm_tol = x - xtol ib = block_head loop while ib # 0 iz = b_zone(ib) loop while iz # 0 if z_x(iz) < xp_tol then if z_x(iz) > xm_tol then nzgp = 0 igz1 = z_gp(iz,1) ygz1 = gp_y(igz1) if ygz1 < eps then nzgp = nzgp + 1 endif igz2 = z_gp(iz,2) ygz2 = gp_y(igz2) if ygz2 < eps then nzgp = nzgp + 1 endif igz3 = z_gp(iz,3) ygz3 = gp_y(igz3) if ygz3 < eps then nzgp = nzgp + 1 endif if nzgp = 2 then nz = nz + 1 xc = z_x(iz) yc = z_y(iz) ra2 = xc*xc + yc*yc ra = sqrt(ra2) e_val = ra2 * o4c / thtime val1 = exp_int val2 = (1. - exp(-e_val)) / e_val table(tab4,ra) = - (val1 + val2) * o4p table(tab6,ra) = - (val1 - val2) * o4p endif endif endif iz = z_next(iz) end_loop

301 THERMAL ANALYSIS 3-57 ib = b_next(ib) end_loop ns = ns + 1 end_loop end ; save line_fish.sav ; num_soltu ana_soltu num_solst ana_solst label tab 1 Temp. 1 Year - UDEC label tab 2 Temp. 1 Year - Analytic label table 3 Radial Stress - UDEC label table 4 Radial Stress - Anal label table 5 Tangential Stress - UDEC label table 6 Tangential Stress - Anal label table 7 Radial Displacement - UDEC label table 8 Radial Displacement - Anal ; save line_compare.sav ; pl hold tab 1 cross 2 pl hold tab 3 cross 4 5 cross 6 pl hold tab 7 cross 8 ret

302 3-58 Special Features Structures/Fluid Flow/Thermal/Dynamics Example 3.5 Exponential integral function ; --- Exponential integral E1(e_val) --- ; Input : e_val ; def exp_int if e_val < 0.0 then ii=out( Argument of Exponential function must be positive ) exit end_if if e_val = 0.0 then exp_int = 1.e12 exit endif if e_val < 1. then e_e1 = (( * e_val ) * e_val ) * e_val e_e1 = ((e_e ) * e_val ) * e_val exp_int = e_e ln(e_val) else e_e1 = e_val * ( e_val) e_e1 = e_e1 / ( e_val * ( e_val)) exp_int = e_e1 * exp(-e_val) / e_val end_if end The results for temperature, radial displacement, and radial and tangential stress distributions at 1 year are presented in the table plots in Figures 3.24 through The differences between numerical and analytical values are generally within 10%. Improved agreement can be expected as the block rounding length is decreased. The outer boundary also has an influence on the numerical results farther from the source.

303 THERMAL ANALYSIS 3-59 JOB TITLE : Infinite line heat source in infinite elastic medium UDEC (Version 5.00) (e-001) 4.50 LEGEND 8-Feb :23:04 cycle 9000 time 8.874E-01 sec thermal time = 3.110E+07 sec table plot Temp. 1 Year - UDEC X X X Temp. 1 Year - Analytic Vs. 1.00E+00<X value> 4.88E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+002) Figure 3.24 Temperature distribution at 1 year JOB TITLE : Infinite line heat source in infinite elastic medium UDEC (Version 5.00) (e-001) 8.00 LEGEND 8-Feb :23:04 cycle 9000 time 8.874E-01 sec thermal time = 3.110E+07 sec table plot Radial Displacement - UDEC X X X Radial Displacement - Anal Vs. 1.00E+00<X value> 4.88E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+002) Figure 3.25 Radial displacement distribution at 1 year

304 3-60 Special Features Structures/Fluid Flow/Thermal/Dynamics JOB TITLE : Infinite line heat source in infinite elastic medium UDEC (Version 5.00) (e-001) 0.50 LEGEND 8-Feb :23:04 cycle 9000 time 8.874E-01 sec thermal time = 3.110E+07 sec table plot Radial Stress - UDEC X X X Radial Stress - Anal Tangential Stress - UDEC X X X Tangential Stress - Anal Vs. 1.54E+00<X value> 4.64E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+002) Figure 3.26 Radial and tangential stress distributions at 1 year

305 THERMAL ANALYSIS References Carslaw, H. S., and J. C. Jaeger. Conduction of Heat in Solids. London: Oxford University Press (1959). Karlekar, B. V., and R. M. Desmond. Heat Transfer: Solutions Manual, 2nd ed. West Publishing Co. (May 1982). Nowacki, W. Thermoelasticity. Addison-Wesley (1962). Ozisik, M. N. Heat Conduction. John Wiley & Sons (1980). Schneider, P. J. Conduction Heat Transfer. Cambridge, Mass.: Addison-Wesley (1955). Timoshenko, S. P., and J. N. Goodier. Theory of Elasticity. New York: McGraw-Hill (1970).

306 3-62 Special Features Structures/Fluid Flow/Thermal/Dynamics

307 DYNAMIC ANALYSIS DYNAMIC ANALYSIS 4.1 Overview Dynamic analysis in UDEC permits two-dimensional, plane-strain or plane-stress, fully dynamic analysis. The calculation is based on the explicit finite difference scheme (as discussed in Section 1 in Theory and Background) to solve the full equations of motion, using real rigid-block masses, or lumped gridpoint masses derived from the real density of surrounding zones (rather than scaled masses used for static solution). Background information on the dynamic formulation of the fully nonlinear method implemented in UDEC is provided. (See Section 4.2.) The dynamic formulation can be coupled to the structural element model, thus permitting analysis of rock-structure interaction brought about by ground shaking. The dynamic feature can also be coupled to the model for fluid flow in joints. This permits, for example, analyses of the effect of dynamic loading of saturated joints. The dynamic model can likewise be coupled to the optional thermal model in order to calculate the combined effect of thermal and dynamic loading. The dynamic facility expands UDEC s analytic capability to a wide range of dynamic problems in disciplines such as earthquake engineering, seismology and mine rockbursts. This volume includes discussions on the various features and considerations associated with the dynamic option in UDEC (i.e., dynamic loading and boundary conditions, wave transmission and mechanical damping). These features are described separately in Section 4.3. The user is strongly encouraged to become familiar with the operation of UDEC for simple mechanical, static problems before attempting to solve problems involving dynamic loading. Dynamic analysis is often very complicated, and requires a considerable amount of insight to interpret correctly. A recommended procedure for conducting dynamic numerical analysis with UDEC is provided in Section 4.4. Validation and example problems illustrating the application of the dynamic model are provided in Section 4.5.* * The data files in this section are stored in the directory ITASCA\UDEC500\Datafiles\Dynamic with the extension.dat. A project file is also provided for each example. In order to run an example and compare the results to plots in this section, open a project file in the GIIC by clicking on the File / Open Project menu item and selecting the project file name (with extension.prj ). Click on the Project Options icon at the top of the Project Tree Record, select Rebuild unsaved states, and the example data file will be run, and plots created.

308 4-2 Special Features Structures/Fluid Flow/Thermal/Dynamics 4.2 Dynamic Formulation The finite difference formulation is identical to that described in Section 1 in Theory and Background, except that real masses are used at rigid-block centroids or deformable-block gridpoints instead of the scaled inertial masses used for optimum convergence in the static solution scheme. For deformable blocks, each triangular zone contributes one-third of its mass (computed from zone density and area) to each of the three associated gridpoints. In finite-element terminology, UDEC uses lumped masses and a diagonal mass matrix. The calculation of critical timestep is identical to that given in Section in Theory and Background. If stiffness-proportional damping is used (see Section ), the timestep must be reduced, for stability. Belytschko (1983) provides a formula for critical timestep, t β, that includes the effect of stiffness-proportional damping: t β = { 2 ω max } ( 1 + λ 2 λ ) (4.1) where ω max is the highest eigenfrequency of the system, and λ is the fraction of critical damping at this frequency. Both ω max and λ are estimated in UDEC, since an eigenvalue solution is not performed. The estimates are given ω max = 2 t d (4.2) λ = 0.4 β t d (4.3) β = ξ min /ω min (4.4) where ξ min and ω min are the damping fraction and angular frequency specified for Rayleigh damping (see Section ), and t d is the timestep for dynamic runs when no stiffness-proportional damping is used. The resulting value of t β is used as the dynamic timestep if stiffness-proportional damping is in operation.

309 DYNAMIC ANALYSIS Dynamic Modeling Considerations There are three aspects that the user should consider when preparing a UDEC model for a dynamic analysis: (1) dynamic loading and boundary conditions; (2) wave transmission through the model; and (3) mechanical damping. This section provides guidance on addressing each aspect when preparing a UDEC data file for dynamic analysis. Section 4.4 illustrates the use of most of the features discussed here Dynamic Loading and Boundary Conditions UDEC models a region of jointed material subjected to external and/or internal dynamic loading by applying a dynamic input boundary condition either at the model boundary or to internal blocks. Wave reflections at model boundaries are minimized by specifying either quiet (viscous) or freefield boundary conditions. The types of dynamic loading and boundary conditions are shown schematically in Figure 4.1; each condition is discussed in the following sections. structure free field quiet boundary internal dynamic input quiet boundary free field quiet boundary external dynamic input (stress or force only) (a) Flexible base structure free field quiet boundary internal dynamic input quiet boundary free field external dynamic input (velocity) (a) Rigid base Figure 4.1 Types of dynamic loading and boundary conditions in UDEC

310 4-4 Special Features Structures/Fluid Flow/Thermal/Dynamics Application of Dynamic Input In UDEC, the dynamic input can be applied in one of four ways: (a) a velocity history; (b) a stress (or pressure) history; (c) a force history; or (d) a fluid pressure history within joints. Dynamic input is usually applied to the boundaries of deformable-block models with the BOUNDARY command. Forces or pressures can also be applied to interior blocks by using the BOUNDARY interior command. Fluid pressures within joint domains are applied with the PFIX command. The history function for the input is treated as a multiplier on the value specified with the BOUNDARY or PFIX command. The history multiplier is assigned with the history keyword, and can be in one of four forms: (1) an harmonic function defined by the sine or cosine keyword; (2) a table defined by the TABLE command; (3) a history input by the BOUNDARY hread command; or (4) a FISH function. A sine or cosine wave is applied with a specified frequency and time period. With TABLE input, the multiplier values and corresponding time values are entered as individual pairs of numbers in the specified table; the first number of each pair is assumed to be a value of dynamic time. The time intervals between successive table entries need not be the same for all entries. When using the BOUNDARY hread command to input the history multiplier, the values stored in the specified history are assumed to be spaced at constant intervals of dynamic time. The interval is contained in the input data file; the file is associated with a particular history number, n. The history is then applied with the BOUNDARY history n command. If a FISH function is used to provide the multiplier, the function must access dynamic time within the function, using the UDEC scalar variable time, and compute a multiplier value that corresponds to this time. Example 4.4 provides an example of dynamic loading derived from a FISH function. Dynamic input can be applied either in the x- ory-directions corresponding to the xy-axes for the model, or in the normal and shear directions to the model boundary. Histories can only be specified for input in the x- and y-directions.

311 DYNAMIC ANALYSIS 4-5 Dynamic input can also be applied for rigid block models. Velocities are applied to rigid blocks by first fixing the block with the FIX command, and then specifying the velocity components with the UDEC model variable b xvel or b yvel. Loads can be applied to rigid blocks with the LOAD command. History functions for the velocities or loads are specified via FISH. An application of a velocity history is illustrated in Example 4.4. One restriction when applying velocity input to model boundaries is that this boundary condition cannot be applied along the same boundary as a quiet (viscous) boundary condition (compare Figure 4.1(a) to Figure 4.1(b)). To overcome this, a stress boundary condition can be used instead (i.e., a velocity record can be transformed into a stress record and applied to a quiet boundary). A velocity wave may be converted to a stress wave using or σ n = 2(ρ C p )v n (4.5) where σ n = applied normal stress; σ s = applied shear stress; ρ = mass density; C p = speed of p-wave propagation through medium; C s = speed of s-wave propagation through medium; v n = input normal particle velocity; and v s = input shear particle velocity. σ s = 2(ρ C s )v s (4.6) C p is given by C p = K + 4G/3 ρ (4.7) and C s is given by C s = G/ρ (4.8) The formulae assume plane-wave conditions. The factor of two in Eqs. (4.5) and (4.6) accounts for the fact that the applied stress must be doubled to overcome the effect of the viscous boundary. The formulation is similar to that of Joyner and Chen (1975). Note that, in this case, a velocity history obtained at the boundary may be different than that from the original velocity record, because of the one-dimensional approximations of Eqs. (4.5) and (4.6).

312 4-6 Special Features Structures/Fluid Flow/Thermal/Dynamics To illustrate wave input at a quiet boundary, consider Example 4.1, in which a pulse is applied as a stress history to the bottom of a vertical, 50-m high column. The bottom of the column is declared quiet in the horizontal direction, and the top is free. The properties are chosen such that the shear wave speed is 100 m/sec, and the product, ρc s,is10 5. The amplitude of the stress pulse is set, therefore, to , according to Eq. (4.6), in order to generate a velocity amplitude of 1 m/sec in the column. Figure 4.2 shows time histories of x-velocity at the base, middle and top of the column; the amplitude of the outgoing wave is seen to be 1 m/sec, as expected. The first three pulses in Figure 4.2 correspond, in order, to the outgoing waves at the base, middle and top. The final two pulses correspond to waves reflected from the free surface, measured at the middle and base, respectively. The velocity-doubling effect of a free surface (as well as the lack of waves after a time of about 1.3 seconds) can be seen, which confirms that the quiet base is working correctly. (See Section ) The doubling effect associated with a free surface is described in texts on elastodynamics (e.g., Graff 1991). Example 4.1 Shear wave propagation in a vertical column new ;file: wave.dat ; Shear wave propagation in a vertical column round 5E-2 edge 0.1 block 0,0 0,50 1,50 1,0 gen quad 0.25 group zone block zone model elastic density 1E3 bulk 2E7 shear 1E7 range group block def wave if time > 1.0 / freq wave = 0.0 else wave = 0.5 * (1.0 - cos(2.0*pi*freq * time)) endif end set freq=4.0 wave boundary yvelocity 0 boundary xvisc ff_bulk=2.0e7 ff_shear=1.0e7 ff_density= range 0,1 & -0.1,0.1 boundary stress 0.0, ,0.0 history=wave range 0,1-0.1,0.1 history xvelocity 0.5,0.0 history xvelocity 0.5,25.0 history xvelocity 0.5,50.0 damping save wave1.sav cycle time 1.7 save wave2.sav

313 DYNAMIC ANALYSIS 4-7 JOB TITLE : Shear Wave Propagation UDEC (Version 5.00) 2.40 LEGEND 16-Dec :59:16 cycle 5634 time 1.701E+00 sec history plot Y-axis: 1 - x-velocity 2 - x-velocity 3 - x-velocity X-axis: Time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 4.2 Primary and reflected waves in a bar: stress input through a quiet boundary Baseline Correction If a raw acceleration or velocity record from a site is used as a time history, the UDEC model may exhibit continuing velocity or residual displacements after the motion has finished. This arises from the fact that the integral of the complete time history may not be zero. For example, the idealized velocity waveform in Figure 4.3(a) may produce the displacement waveform in Figure 4.3(b) when integrated. The process of baseline correction should be performed, although the physics of the UDEC simulation usually will not be affected if it is not done. It is possible to determine a low frequency wave (for example, Figure 4.3(c)) which, when added to the original history, produces a final displacement which is zero (Figure 4.3(d)). The low frequency wave in Figure 4.3(c) can be a polynomial or periodic function, with free parameters that are adjusted to give the desired results. Baseline correction usually applies only to complex waveforms derived, for example, from field measurements. When using a simple, synthetic waveform, it is easy to arrange the process of generating the synthetic waveform to ensure that the final displacement is zero. Normally, in seismic analysis, the input wave is an acceleration record. A baseline-correction procedure can be used to force both the final velocity and displacement to be zero. Earthquake engineering texts should be consulted for standard baseline correction procedures.

314 4-8 Special Features Structures/Fluid Flow/Thermal/Dynamics velocity (a) velocity history time displacement (b) displacement history time velocity time (c) low frequency velocity wave displacement time (d) resultant displacement history Figure 4.3 The baseline correction process An alternative to baseline correction of the input record is to apply a displacement shift at the end of the calculation if there is a residual displacement of the entire model. This can be done by applying a fixed velocity to the mesh to reduce the residual displacement to zero. This action will not affect the mechanics of the deformation of the model. Computer codes to perform baseline corrections are available from several Internet sites (e.g., Quiet Boundaries The modeling of geomechanics problems involves media which, at the scale of the analysis, are better represented as unbounded. Deep underground excavations are normally assumed to be surrounded by an infinite medium, while surface and near-surface structures are assumed to lie on a half-space. Numerical methods relying on the discretization of a finite region of space require that appropriate conditions be enforced at the artificial numerical boundaries. In static analyses, fixed or elastic boundaries (e.g., represented by boundary-element techniques) can be realistically placed at some distance from the region of interest. In dynamic problems, however, such boundary conditions cause the reflection of outward propagating waves back into the model, and do not allow the necessary energy radiation. The use of a larger model can minimize the problem, since material damping will absorb most of the energy in the waves reflected from distant boundaries. However,

315 DYNAMIC ANALYSIS 4-9 this solution leads to a large computational burden. The alternative is to use quiet (or absorbing) boundaries. Several formulations have been proposed. The viscous boundary developed by Lysmer and Kuhlemeyer (1969) is used in UDEC. It is based on the use of independent dashpots in the normal and shear directions at the model boundaries. The method is almost completely effective at absorbing body waves approaching the boundary at angles of incidence greater than 30. For lower angles of incidence, or for surface waves, there is still energy absorption, but it is not perfect. However, the scheme has the advantage that it operates in the time domain. Its effectiveness has been demonstrated in both finite-element and finite-difference models (Kunar et al. 1977). A variation of the technique proposed by White et al. (1977) is also widely used. More efficient energy absorption (particularly in the case of Rayleigh waves) requires the use of frequency-dependent elements, which can only be used in frequency-domain analyses (e.g., Lysmer and Waas 1972). These are usually termed consistent boundaries, and involve the calculation of dynamic stiffness matrices coupling all the boundary degrees-of-freedom. Boundary element methods may be used to derive these matrices (e.g., Wolf 1985). A comparative study of the performance of different types of elementary, viscous and consistent boundaries was documented by Roesset and Ettouney (1977). A different procedure to obtain efficient absorbing boundaries for use in time domain studies was proposed by Cundall et al. (1978). It is based on the superposition of solutions with stress and velocity boundaries in such a way that reflections are canceled. In practice, it requires adding the results of two parallel, overlapping grids in a narrow region adjacent to the boundary. This method has been shown to provide effective energy absorption, but is difficult to implement for a blocky system with complex geometry, and thus is not used in UDEC. The quiet-boundary scheme proposed by Lysmer and Kuhlemeyer (1969) involves dashpots attached independently to the boundary in the normal and shear directions. The dashpots provide viscous normal and shear tractions given by t n = ρc p v n (4.9) t s = ρc s v s (4.10) where v n and v s ρ C p and C s are the normal and shear components of the velocity at the boundary; is the mass density; and are the p- and s-wave velocities. These viscous terms can be introduced directly into the equations of motion of the gridpoints lying on the boundary. A different approach, however, was implemented in UDEC, whereby the tractions t n and t s are calculated and applied at every timestep in the same way boundary loads are applied. This is more convenient than the former approach, and tests have shown that the implementation is equally effective. The only potential problem concerns numerical stability, because the viscous forces are calculated from velocities lagging by half a timestep. In practical analyses to date, no reduction of timestep has been required by the use of the nonreflecting boundaries. Timestep restrictions demanded by small zones are usually more important.

316 4-10 Special Features Structures/Fluid Flow/Thermal/Dynamics Dynamic analysis starts from some in-situ equilibrium condition. If a velocity boundary is used to provide the static stress state, this boundary condition can be replaced by quiet boundaries; the boundary reaction forces will be automatically calculated and maintained throughout the dynamic loading phase. However, be careful to avoid changes in static loading during the dynamic phase. For example, if a tunnel is excavated after quiet boundaries have been specified on the bottom boundary, the whole model will start to move upward. This is because the total gravity force no longer balances the total reaction force at the bottom (calculated when the boundary was changed to a quiet one). If a stress boundary condition is applied for the static solution, a stress boundary condition of opposite sign must also be applied over the same boundary when the quiet boundary is applied for the dynamic phase. This will allow the correct reaction forces to be in place at the boundary for the dynamic calculation. Quiet boundary conditions can be assigned to deformable blocks in the x- and y-directions. The boundary conditions are applied with the BOUNDARY xvisc and BOUNDARY yvisc commands. The command BOUNDARY mat assigns a material property number to the far-field properties used for Eqs. (4.9) and (4.10). A quiet boundary can also be applied to a rigid block model by first creating deformable blocks at the boundary (see Example 4.4) Free-Field Boundaries Seismic analysis by numerical techniques of surface structures such as dams requires the discretization of a region of the material adjacent to the foundation. The seismic input is normally represented by plane waves propagating upward through the underlying material. The boundary conditions at the sides of the model must account for the free-field motion which would exist in the absence of the structure. In some cases, elementary lateral boundaries may be sufficient. For example, if only a shear wave were applied on AC (shown in Figure 4.4), it would be possible to fix the boundary along AB and CD in the y-direction only (see the example in Section 4.5.2). These boundaries should be placed at sufficient distances to minimize wave reflections and achieve free-field conditions. For soils with high material damping, this condition can be obtained with a relatively small distance (Seed et al. 1975). However, when the material damping is low, the required distance may lead to an impractical model. An alternative procedure is to enforce the free-field motion in such a way that boundaries retain their nonreflecting properties (i.e., outward waves originating from the structure are properly absorbed). This approach was used in the continuum finite-difference code NESSI (Cundall et al. 1980). A technique of this type was developed for UDEC. It involves the execution of a one-dimensional free-field calculation in parallel with the main system analysis. The lateral boundaries are coupled to the free-field grid by viscous dashpots to simulate a quiet boundary (see Figure 4.4), and the unbalanced forces from the free-field grid are applied to the deformable-block boundary at the boundary gridpoints. Both conditions are expressed in Eqs. (4.11) and (4.12), which apply to the left-hand boundary. Similar expressions may be written for the right-hand boundary. F x = ρc p (v m x vff x ) + σ ff xx S y (4.11) F y = ρc s (v m y vff y ) + σ ff xy S y (4.12)

317 DYNAMIC ANALYSIS 4-11 where ρ = density of material along vertical model boundary; C p = p-wave speed at the left-hand boundary; C s = s-wave speed at the left-hand boundary; S y = mean vertical zone size at boundary gridpoint; vx m = x-velocity of gridpoint in deformable block at left boundary; vy m = y-velocity of gridpoint in deformable block at left boundary; = x-velocity of gridpoint in left free field; v ff x v ff y σ ff xx σ ff xy = y-velocity of gridpoint in left free field; = mean horizontal free-field stress at gridpoint; and = mean free-field shear stress at gridpoint. In this way, plane waves propagating upward suffer no distortion at the boundary because the freefield grid supplies conditions that are identical to those in an infinite model. If the deformable-block model is uniform, and there is no surface structure, the lateral dashpots are not exercised because the free-field grid executes the same motion as the main model. However, if the main model motion differs from that of the free field (due, say, to a surface structure that radiates secondary waves), then the dashpots act to absorb energy in a manner similar to that of quiet boundaries. B D free field free field A C seismic wave Figure 4.4 Model for seismic analysis of surface structures and free-field mesh A free-field boundary is invoked with the BOUNDARY ffield command. The free field is created, and in-situ conditions prior to the dynamic analysis are assigned to the free field. Note that the free-field boundary conditions require that lateral boundaries of the main model must be vertical and straight. The free field is also connected to the main model with the BOUNDARY ffield command. The free-field grid can only be connected to deformable blocks. The BOUNDARY ffield command must be given prior to assigning the boundary that specifies the dynamic loading. The free-field model consists of a one-dimensional column of unit width, simulating the behavior of the extended medium. An explicit finite-difference method was selected for the model. The

318 4-12 Special Features Structures/Fluid Flow/Thermal/Dynamics height of the free field equals the length of the lateral boundaries. It is discretized into n elements corresponding to the zones along the lateral boundaries of the UDEC model. Element masses are lumped at the n + 1 gridpoints. A linear variation of the displacement field is assumed within each element; the elements are, therefore, in a state of uniform strain (and stress). The application of the free-field boundary is illustrated in Example 4.2. A shear-stress wave is applied to the base of the model. Figure 4.5 shows the resulting x-velocity at the top of the model at different locations in the free field and the main block. Example 4.2 Shear wave loading of a model with free-field boundaries new ;File: ffield.dat round 1.6E-2 block 0,0 0,10 16,10 16,0 gen quad 1.0 group zone block zone model elastic density 2.5E-3 bulk 6.667E4 shear 4E4 range group & block set gravity= boundary xvelocity 0 range -0.1, ,10.1 boundary xvelocity 0 range 15.9, ,10.1 boundary yvelocity 0 range -0.1, ,0.1 solve ratio 1.0E-5 save ffield1.sav boundary ffield boundary xvisc ff_bulk= ff_shear= ff_density= range & -0.1, ,0.1 boundary yvisc ff_bulk= ff_shear= ff_density= range & -0.1, ,0.1 def wave wave = 0.5 * (1.0 - cos(2*pi*time/period)) end set period boundary stress 0.0,-1.0,0.0 history=wave range -0.1, ,0.1 reset time history xvelocity 0.0,10.0 history xvelocity 7.6,10.0 history xvelocity 16.0,10.0 history ffxv 10 2 damp cycle time 0.02 save ffield2.sav

319 DYNAMIC ANALYSIS 4-13 JOB TITLE : UDEC (Version 5.00) (e-001) 1.20 LEGEND 24-Jan :54:41 cycle 1741 time 2.005E-02 sec history plot Y-axis: 1 - x-velocity 2 - x-velocity 3 - x-velocity X-axis: Number of cycles Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+003) Figure 4.5 x-velocity histories at top of model with free-field boundaries Example applications of the free-field boundary are also given in Section 3.6 in the User s Guide and Section 4 in the Example Applications.

320 4-14 Special Features Structures/Fluid Flow/Thermal/Dynamics Deconvolution and Selection of Dynamic Boundary Conditions Design earthquake ground motions developed for seismic analyses are usually provided as outcrop motions, often rock outcrop motions.* However, for UDEC analyses, seismic input must be applied at the base of the model rather than at the ground surface, as illustrated in Figure 4.6. The question then arises, What input motion should be applied at the base of a UDEC model in order to properly simulate the design motion? The appropriate input motion at depth can be computed through a deconvolution analysis using a 1D wave propagation code such as the equivalent-linear program SHAKE. This seemingly simple analysis is often the subject of considerable confusion resulting in improper ground motion input for UDEC models. The application of SHAKE for adapting design earthquake motions for UDEC input is described. Input of an earthquake motion into UDEC is typically done using one of two base types: 1. A rigid base, where an acceleration-time history is specified at the base of the UDEC model. 2. A compliant base, where a quiet (absorbing) boundary is used at the base of the UDEC model. Figure 4.6 Seismic input to UDEC For a rigid base, a time history of acceleration (or velocity or displacement) is specified for deformable-block gridpoints (or rigid-block centroids) along the base of the model. While simple to use, a potential drawback of a rigid base is that the motion at the base of the model is completely prescribed. Hence, the base acts as if it were a fixed displacement boundary reflecting downwardpropagating waves back into the model. Thus, a rigid base is not an appropriate boundary for * This section is abstracted with permission from the publication by Mejia and Dawson (2006).

321 DYNAMIC ANALYSIS 4-15 general application unless a large dynamic impedance contrast is meant to be simulated at the base (e.g. low velocity sediments over high velocity bedrock). For a compliant base simulation, a quiet boundary is specified along the base of the UDEC model. See Section for a description of quiet boundaries. Note that if a history of acceleration is recorded at a point along the quiet base, it will not necessarily match the input history. The input stress-time history specifies the upward-propagating wave motion in to the UDEC model, but the actual motion at the base will be the superposition of the upward motion and the downward motion reflected back from the UDEC model. SHAKE (Schnabel et al. 1972) is a widely used 1D wave propagation code for site response analysis. SHAKE computes the vertical propagation of shear waves through a profile of horizontal visco-elastic layers. Within each layer, the solution to the wave equation can be expressed as the sum of an upward-propagating wave train and a downward-propagating wave train. The SHAKE solution is formulated in terms of these upward- and downward-propagating motions within each layer, as illustrated in Figure 4.7: Figure 4.7 Layered system analyzed by SHAKE (layer properties are shear modulus, G, density, ρ, and damping fraction, ζ ) The relation between waves in one layer and waves in an adjacent layer can be solved by enforcing the continuity of stresses and displacements at the interface between the layers. These well-known relations for reflected and transmitted waves at the interface between two elastic materials (Kolsky 1963) can be expressed in terms of recursion formulas. In this way, the upward- and downwardpropagating motions in one layer can be computed from the upward and downward motions in a neighboring layer. To satisfy the zero shear stress condition at the free surface, the upward- and downward-propagating motions in the top layer must be equal. Starting at the top layer, repeated use of the recursion formulas allows the determination of a transfer function between the motions in any two layers of the system. Thus, if the motion is specified at one layer in the system, the motion at any other layer can be computed. SHAKE input and output is not in terms of the upward- and downward-propagating wave trains, but in terms of the motions at a) the boundary between two layers, referred to as a within motion; or

322 4-16 Special Features Structures/Fluid Flow/Thermal/Dynamics b) at a free surface, referred to as an outcrop motion. The within motion is the superposition of the upward- and downward-propagating wave trains. The outcrop motion is the motion that would occur at a free surface at that location. Hence the outcrop motion is simply twice the upward-propagating wave-train motion. If needed, the upward-propagating motion can be computed by taking half the outcrop motion. At any point, the downward-propagating motion can then be computed by subtracting the upward-propagating motion from the within motion. The SHAKE solution is in the frequency domain, with conversion to and from the time-domain performed with a Fourier transform. The deconvolution analysis discussed below illustrates the application of SHAKE for a linear, elastic case. SHAKE can also address nonlinear soil behavior approximately, through the equivalent linear approach. Analyses are run iteratively to obtain shear modulus and damping values for each layer that are compatible with the computed effective strain for the layer. Deconvolution for a Rigid Base The deconvolution procedure for a rigid base is illustrated in Figure 4.8 for a two-dimensional FLAC simulation. The same procedure also applies to UDEC. The goal is to determine the appropriate base input motion to FLAC, such that the target design motion is recovered at the top surface of the FLAC model. The profile modeled consists of three 20-m thick elastic layers with shear wave velocities and densities as shown in the figure. The SHAKE model includes the three elastic layers and an elastic half-space with the same properties as the bottom layer. The FLAC model consists of a column of linear elastic elements. The target earthquake is input at the top of the SHAKE column as an outcrop motion. Then, the motion at the top of the half-space is extracted as a within motion, and is applied as an acceleration-time history to the base of the FLAC model. Mejia and Dawson (2006) show that the resulting acceleration at the surface of the FLAC model is virtually identical to the target motion. The SHAKE within motion is appropriate for rigid-base input because (as described above) the within motion is the actual motion at that location, the superposition of the upward- and downward-propagating waves. Deconvolution for a Compliant Base The deconvolution procedure for a compliant base is illustrated in Figure 4.9 for a FLAC simulation. Again, the same procedure applies to UDEC. The SHAKE and FLAC models are identical to those for the rigid body exercise, except that a quiet boundary is applied to the base of the FLAC mesh. For application through a quiet base, the upward-propagating wave motion (1/2 the outcrop motion) is extracted from SHAKE at the top of the half-space. This acceleration-time history is integrated to obtain a velocity, which is then converted to a stress history using Eq. (4.6). Again, the resulting acceleration at the surface of the FLAC model is shown by Mejia and Dawson (2006) to be virtually identical to the target motion. As an additional check of the computed accelerations, they also show that the response spectra for both the compliant-base and rigid-base cases closely match the response spectra of the target motion.

323 DYNAMIC ANALYSIS 4-17 Figure 4.8 Deconvolution procedure for a rigid base (after Mejia and Dawson 2006) Figure 4.9 Deconvolution procedure for a compliant base (after Mejia and Dawson 2006)

324 4-18 Special Features Structures/Fluid Flow/Thermal/Dynamics Although useful for illustrating the basic ideas behind deconvolution, the previous example is not the typical case encountered in practice. The situation shown in Figure 4.10, where one or more soil layers (expected to behave nonlinearly) overlay bedrock (assumed to behave linearly), is more common. A FLAC or UDEC model for this case will usually include the soil layers and an elastic base of bedrock. To compute the correct UDEC compliant base input, a SHAKE model is constructed as shown in the figure. The SHAKE model includes a bedrock layer equal in thickness to the elastic base of the UDEC model, and an underlying elastic half-space with bedrock properties. The target motion is input to the SHAKE model as an outcrop motion at the top of the bedrock (point A). Designating this motion as outcrop means that the upward-propagating wave motion in the layer directly below point A will be set equal to 1/2 the target motion. The upward-propagating motion for input to UDEC is extracted at Point B as 1/2 the outcrop motion. For the compliant-base case there is actually no need to include the soil layers in the SHAKE model, as these will have no effect on the upward-propagating wave train between points A and B. In fact, for this simple case, it is not really necessary to perform a formal deconvolution analysis, as the upward-propagating motion at point B will be almost identical to that at point A. Apart from an offset in time, the only differences will be due to material damping between the two points, which will generally be small for bedrock. Thus, for this very common situation, the correct input motion for UDEC is simply 1/2 of the target motion. (Note that the upward-propagating wave motion must be converted to a stress-time history using Eq. (4.6), which includes a factor of 2 to account for the stress absorbed by the viscous dashpots.) For a rigid-base analysis, the within motion at point B is required. Since this within motion incorporates downward-propagating waves reflected off the ground surface, the nonlinear soil layers must be included in the SHAKE model. However, soil nonlinearity will be modeled quite differently in UDEC and SHAKE. Thus, it is difficult to compute the appropriate UDEC input motion for a rigid-base analysis. Another typical case encountered in practice is illustrated in Figure Here, the soil profile is deep, and rather than extending the UDEC model all the way down to bedrock, the base of the model ends within the soil profile. Note that the model must be extended to a depth below which the soil response is essentially linear. Again, the design motion is input at the top of the bedrock (point A) as an outcrop motion, and the upward-propagating motion for input to UDEC is extracted at point B. As in the previous example, for a compliant-base analysis there is no need to include the soil layers above point B in the SHAKE model. These layers have no effect on the upward-propagating motion between points A and B. Unlike the previous case, the upward-propagating motion can be quite different at points A and B, depending on the impedance contrast between the bedrock and linear soil layer. Thus, it is not appropriate to skip the deconvolution analysis and use the target motion directly. A rigid base is only appropriate for cases with a large impedance contrast at the base of the model. However, the use of SHAKE to compute the required input motion for a rigid base of a UDEC model leads to a good match between the target surface motion and the surface motion computed by UDEC, only for a model that exhibits a low level of nonlinearity. The input motion already contains the effect of all layers above the base, because it contains the downward-propagating wave.

325 DYNAMIC ANALYSIS 4-19 Figure 4.10 Compliant-base deconvolution procedure for a typical case (after Mejia and Dawson 2006) Figure 4.11 Compliant-base deconvolution procedure for another typical case (after Mejia and Dawson 2006)

326 4-20 Special Features Structures/Fluid Flow/Thermal/Dynamics A different approach must be taken if a UDEC model with a rigid base is used to simulate more realistic systems (e.g., sites that exhibit strong nonlinearity, or the effect of a surface or embedded structure). In the first case, the real nonlinear response is not accounted for by SHAKE in its estimate of base motion. In the second case, secondary waves from the structure will be reflected from the rigid base, causing artificial resonance effects. A compliant base is almost always the preferred option because downward-propagating waves are absorbed. In this case, the quiet-base condition is selected, and only the upward-propagating wave from SHAKE is used to compute the input stress history. By using the upward-propagating wave only at a quiet UDEC base, no assumptions need to be made about secondary waves generated by internal nonlinearities or structures within the grid, because the incoming wave is unaffected by these; the outgoing wave is absorbed by the compliant base. Although the presence of reflections from a rigid base is not always obvious in complex nonlinear UDEC analyses, they can have a major impact on analysis results, especially when cyclicdegradation or liquefaction-soil models are employed. Mejia and Dawson (2006) present examples from two-dimensional FLAC simulations that illustrate the nonphysical wave reflections calculated in models with a rigid base. One example, shown in Figure 4.12, demonstrates the difficulty with a rigid boundary. The nonphysical oscillations that result from a rigid base are shown by comparison to results for a compliant base in Figure The inputs in both cases (rigid and compliant) were derived by deconvoluting the same surface motion. Figure 4.12 Embankment analyzed with a rigid and compliant base (after Mejia and Dawson 2006) Figure 4.13 Computed accelerations at crest of embankment (after Mejia and Dawson 2006)