fluid mechanical interaction

FLAC3DFast Lagrangian Analysis of Continua

in 3 DimensionsFluid-Mechanical Interaction

©2009

Itasca Consulting Group Inc. Phone: (1) 612-371-4711

Mill Place Fax: (1) 612·371·4717

111 Third Avenue South, Suite 450 E-Mail: [email protected]

Minneapolis, Minnesota 55401 USA Web: www.itascacg.com

First Edition (FLAC3D Version 2.1) April 2002

First Revision (FLAC3D Version 2.1) June 2003

Second Edition (FLAC3D Version 3.0) September 2005

First Revision December 2005*

Third Edition (FLAC3D Version 3.1) December 2006

Fourth Edition (FLAC3D Version 4.0) December 2009

* Please see the errata page in the \Manuals\Flac3d400 folder.

Fluid-Mechanical Interaction Contents - 1

TABLE OF CONTENTS

1 FLUID-MECHANICAL INTERACTION – SINGLE PHASE FLUID1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 1

1.2 FLAC3D Fluid-Thermal-Mechanical Formulation – Mathematical Description . 1 - 41.2.1 Conventions and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 41.2.2 Governing Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 61.2.3 Fluid Flow Boundary and Initial Conditions in FLAC3D . . . . . . . . . . . . . 1 - 9

1.3 Numerical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 101.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 101.3.2 Saturated Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 11

1.3.2.1 Finite-Difference Approximation to Space Derivatives . . . . . . . 1 - 111.3.2.2 Nodal Formulation of the Mass Balance Equation . . . . . . . . . . . 1 - 121.3.2.3 Explicit Finite-Difference Formulation . . . . . . . . . . . . . . . . . . . . . 1 - 151.3.2.4 Implicit Finite-Difference Formulation . . . . . . . . . . . . . . . . . . . . . 1 - 17

1.3.3 Saturated/Unsaturated Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 201.3.4 Mechanical Timestep and Numerical Stability . . . . . . . . . . . . . . . . . . . . . . 1 - 221.3.5 Total Stress Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 23

1.4 Calculation Modes for Fluid-Mechanical Interaction . . . . . . . . . . . . . . . . . . . . . . . . 1 - 241.4.1 Grid Not Configured for Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 241.4.2 Grid Configured for Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 24

1.5 Properties and Units for Fluid-Flow Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 291.5.1 Permeability Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 291.5.2 Mass Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 301.5.3 Fluid Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 30

1.5.3.1 Biot Coefficient and Biot Modulus . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 301.5.3.2 Fluid Bulk Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 311.5.3.3 Fluid Moduli and Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 321.5.3.4 Fluid Moduli for Drained and Undrained Analyses . . . . . . . . . . 1 - 32

1.5.4 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 331.5.5 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 331.5.6 Undrained Thermal Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 331.5.7 Fluid Tension Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 34

1.6 Fluid-Flow Boundary Conditions, Initial Conditions, Sources and Sinks . . . . . . 1 - 35

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Contents - 2 Fluid-Mechanical Interaction

1.7 Solving Flow-Only and Coupled-Flow Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 371.7.1 Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 371.7.2 Selection of a Modeling Approach for a Fully Coupled Analysis . . . . . 1 - 41

1.7.2.1 Time Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 411.7.2.2 Nature of Imposed Perturbation to the Coupled Process . . . . . . 1 - 421.7.2.3 Stiffness Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 421.7.2.4 Recommended Procedure to Select a Modeling Approach . . . . 1 - 44

1.7.3 Fixed Pore Pressure (Used in Effective Stress Calculation) . . . . . . . . . . . 1 - 471.7.4 Flow-Only Calculation to Establish a Pore-Pressure Distribution . . . . . 1 - 471.7.5 No Flow – Mechanical Generation of Pore Pressure . . . . . . . . . . . . . . . . . 1 - 481.7.6 Coupled Flow and Mechanical Calculations . . . . . . . . . . . . . . . . . . . . . . . . 1 - 50

1.8 Verification Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 591.8.1 Unsteady Groundwater Flow in a Confined Layer . . . . . . . . . . . . . . . . . . . 1 - 591.8.2 One-Dimensional Filling of a Porous Region . . . . . . . . . . . . . . . . . . . . . . . 1 - 671.8.3 Steady-State Fluid Flow with a Free Surface . . . . . . . . . . . . . . . . . . . . . . . 1 - 721.8.4 Spreading of a Groundwater Mound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 841.8.5 One-Dimensional Consolidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 921.8.6 Transient Fluid Flow to a Well in a Shallow Confined Aquifer . . . . . . . . 1 - 1081.8.7 Pressuremeter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 120

1.9 Verification of Concepts, and Modeling Techniques for Specific Applications . 1 - 1341.9.1 Solid Weight, Buoyancy and Seepage Forces . . . . . . . . . . . . . . . . . . . . . . . 1 - 134

1.9.1.1 A Simple Example Illustrating Solid Weight, Buoyancy and Seep-age Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 135

1.9.2 Pore Pressure Initialization and Deformation . . . . . . . . . . . . . . . . . . . . . . . 1 - 1471.9.3 Effect of the Biot Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 153

1.9.3.1 Undrained Oedometer Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 1551.9.3.2 Pore Pressure Generation in a Confined Sample . . . . . . . . . . . . . 1 - 1561.9.3.3 Pore Pressure Generation in an Infinite Layer . . . . . . . . . . . . . . . 1 - 158

1.9.4 Semi-confined Aquifer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 160

1.10 Input Instructions for Fluid-Flow Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 1641.10.1 FLAC3D Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 1641.10.2 FISH Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 174

1.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 175

FLAC3D Version 4.0


TABLES

Table 1.1 Property specification methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 27Table 1.2 Recommended procedure to select a modeling approach for a fully coupled

analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 45

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FIGURES

Figure 1.1 Definitions for principal directions of permeability tensor . . . . . . . . . . . . . . . . 1 - 26Figure 1.2 Instantaneous pore pressures generated under an applied load . . . . . . . . . . . . . 1 - 50Figure 1.3 Pore-pressure distribution at 5000 seconds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 54Figure 1.4 Consolidation response – time histories of footing displacements . . . . . . . . . . 1 - 54Figure 1.5 Swelling displacements near a trench with impermeable surfaces . . . . . . . . . . 1 - 57Figure 1.6 History of pore pressure behind the trench face . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 57Figure 1.7 Displacement histories at the trench crest – vertical (top) and horizontal (bot-

tom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 58Figure 1.8 Confined flow in a soil layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 59Figure 1.9 FLAC3D grid for fluid flow in a confined soil layer . . . . . . . . . . . . . . . . . . . . . . . 1 - 60Figure 1.10 Comparison of excess pore pressures for the explicit-solution algorithm (ana-

lytical values = lines; numerical values = crosses) . . . . . . . . . . . . . . . . . . . . . 1 - 63Figure 1.11 Comparison of excess pore pressures for the implicit-solution algorithm (an-

alytical values = lines; numerical values = crosses) . . . . . . . . . . . . . . . . . . . . 1 - 66Figure 1.12 Location of filling front ( x vs t ) – no gravity

(analytical solution = solid line; numerical values = dashed line) . . . . . . . 1 - 71Figure 1.13 Location of filling front ( x vs t ) – with gravity

(analytical solution = solid line; numerical values = dashed line) . . . . . . . 1 - 72Figure 1.14 Problem geometry and boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 73Figure 1.15 Seepage face solution after Polubarinova-Kochina (1962) . . . . . . . . . . . . . . . . 1 - 74Figure 1.16 Steady-state flow vectors and seepage face solution – Case 1 . . . . . . . . . . . . . . 1 - 79Figure 1.17 Steady-state flow vectors and seepage face solution – Case 2 . . . . . . . . . . . . . . 1 - 79Figure 1.18 Steady-state flow vectors for ATTACHed grid – Case 1 . . . . . . . . . . . . . . . . . . . . 1 - 81Figure 1.19 Steady-state flow vectors for grid with INTERFACE – Case 1 . . . . . . . . . . . . . . 1 - 83Figure 1.20 FLAC3D grid and initial state of saturated column . . . . . . . . . . . . . . . . . . . . . . . 1 - 85Figure 1.21 Saturation contours and analytical mound elevation at t = 0.35 . . . . . . . . . . . . 1 - 90Figure 1.22 Saturation contours and analytical mound elevation at t = 0.45 . . . . . . . . . . . . 1 - 90Figure 1.23 Saturation contours and analytical mound elevation at t = 0.65 . . . . . . . . . . . . 1 - 91Figure 1.24 Saturation contours and analytical mound elevation at t = 0.85 . . . . . . . . . . . . 1 - 91Figure 1.25 Head contours and analytical mound elevation at t = 0.85 . . . . . . . . . . . . . . . . 1 - 92Figure 1.26 One-dimensional consolidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 93Figure 1.27 FLAC3D grid for one-dimensional consolidation . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 95Figure 1.28 Comparison between analytical and numerical values of pore pressure in a

one-dimensional consolidation test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 98Figure 1.29 Comparison between analytical and numerical values for vertical displacement

in a one-dimensional consolidation test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 99Figure 1.30 Evolution of pore pressure, total and effective stresses in a one-dimensional

consolidation test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 99Figure 1.31 Comparison between analytical pore-pressure solutions for two large values

of M in a one-dimensional consolidation test . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 102

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Figure 1.32 Comparison between analytical vertical displacement solutions for two largevalues of M in a one-dimensional consolidation test . . . . . . . . . . . . . . . . . . . 1 - 102

Figure 1.33 Comparison between analytical and numerical values of pore pressure in aone-dimensional consolidation test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 106

Figure 1.34 Comparison between analytical and numerical values for vertical displacementin a one-dimensional consolidation test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 107

Figure 1.35 Evolution of pore pressure, total and effective stresses in a one-dimensionalconsolidation test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 107

Figure 1.36 Flow to a well in a shallow confined aquifer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 108Figure 1.37 FLAC3D grid for a well in a shallow aquifer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 111Figure 1.38 Pore-pressure distribution at 4, 8, 16 and 32 seconds

– coupled solution(analytical values = lines; numerical values = symbols) . . . . . . . . . . . . . . . . 1 - 117

Figure 1.39 Pore-pressure distribution at 4, 8, 16 and 32 seconds– uncoupled solution(analytical values = lines; numerical values = symbols) . . . . . . . . . . . . . . . . 1 - 117

Figure 1.40 Radial, tangential and vertical stress distributions at 32 seconds– coupled solution(analytical values = lines; numerical values = symbols) . . . . . . . . . . . . . . . . 1 - 118

Figure 1.41 Radial, tangential and vertical stress distributions at 32 seconds– uncoupled solution(analytical values = lines; numerical values = symbols) . . . . . . . . . . . . . . . . 1 - 118

Figure 1.42 Radial displacement distribution at 32 seconds– coupled solution(analytical values = line; numerical values = symbols) . . . . . . . . . . . . . . . . . 1 - 119

Figure 1.43 Radial displacement distribution at 32 seconds– uncoupled solution(analytical values = line; numerical values = symbols) . . . . . . . . . . . . . . . . . 1 - 119

Figure 1.44 Cylindrical cavity expansion in pressuremeter test . . . . . . . . . . . . . . . . . . . . . . . 1 - 120Figure 1.45 Domain of FLAC3D simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 122Figure 1.46 FLAC3D grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 122Figure 1.47 Pore-pressure profiles – 300 seconds consolidation . . . . . . . . . . . . . . . . . . . . . . 1 - 124Figure 1.48 Profiles of radial and tangential normal stresses – 300 seconds consolidation 1 - 124Figure 1.49 Pore-pressure profiles – 600 seconds consolidation . . . . . . . . . . . . . . . . . . . . . . 1 - 125Figure 1.50 Profiles of radial and tangential normal stresses – 600 seconds consolidation 1 - 125Figure 1.51 Pore-pressure profile – 600 seconds consolidation

(Mohr-Coulomb material) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 126Figure 1.52 Profiles of radial and tangential normal stresses – 600 seconds consolidation

(Mohr-Coulomb material) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 127Figure 1.53 Vertical stress versus elevation – dry layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 136Figure 1.54 Vertical stress versus elevation – saturated layer . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 137Figure 1.55 Pore pressure contours at steady state – saturated layer . . . . . . . . . . . . . . . . . . . 1 - 138Figure 1.56 Heave of the layer at steady state – saturated layer . . . . . . . . . . . . . . . . . . . . . . . 1 - 139Figure 1.57 Heave of the layer at steady state – seepage force from over-pressured aquifer 1 - 140

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Figure 1.58 Settlement of the layer at steady state – seepage force from under-pressuredaquifer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 142

Figure 1.59 Heave of a soil layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 149Figure 1.60 Decomposition of stresses acting on a porous, elastic rock . . . . . . . . . . . . . . . . 1 - 153Figure 1.61 Shallow semi-confined aquifer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 160Figure 1.62 Pore pressure profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 162

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EXAMPLES

Example 1.1 Adding load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 49Example 1.2 A coupled fluid flow-mechanical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 52Example 1.3 Maintaining equilibrium under time-dependent swelling conditions . . . . . . 1 - 55Example 1.4 Unsteady groundwater flow in a confined layer – explicit solution . . . . . . . . 1 - 61Example 1.5 Unsteady groundwater flow in a confined layer – implicit solution . . . . . . . 1 - 64Example 1.6 1D filling of a porous region – no gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 68Example 1.7 1D filling of a porous region – with gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 69Example 1.8 Steady-state flow through a vertical embankment – Case 1 . . . . . . . . . . . . . . 1 - 75Example 1.9 Steady-state flow through a vertical embankment – Case 2 . . . . . . . . . . . . . . 1 - 77Example 1.10 Steady-state flow through a vertical embankment – Case 1 with ATTACHed

grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 80Example 1.11 Steady-state flow through a vertical embankment – Case 1 with INTERFACE 1 - 82Example 1.12 Spreading of a groundwater mound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 85Example 1.13 Spreading of a groundwater mound – saturation and head plots . . . . . . . . . . 1 - 86Example 1.14 One-dimensional consolidation (coupled) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 96Example 1.15 One-dimensional consolidation (analytical solution) . . . . . . . . . . . . . . . . . . . . 1 - 100Example 1.16 One-dimensional consolidation (uncoupled) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 103Example 1.17 Transient fluid flow to a well in a shallow aquifer . . . . . . . . . . . . . . . . . . . . . . 1 - 113Example 1.18 Exponential integral function – “EXP-INT.FIS” . . . . . . . . . . . . . . . . . . . . . . . . 1 - 116Example 1.19 Pressuremeter test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 127Example 1.20 Generate tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 132Example 1.21 Solid weight, buoyancy and seepage forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 142Example 1.22 Heave of a soil layer, without config fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 149Example 1.23 Heave of a soil layer, with config fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 151Example 1.24 Undrained oedometer test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 155Example 1.25 Pore pressure generation in a confined sample . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 157Example 1.26 Pore pressure generation in an infinite layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 159Example 1.27 Shallow confined aquifer with leaky boundary . . . . . . . . . . . . . . . . . . . . . . . . . 1 - 162

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FLAC3D Version 4.0

FLUID-MECHANICAL INTERACTION – SINGLE PHASE FLUID 1 - 1

1 FLUID-MECHANICAL INTERACTION – SINGLE PHASE FLUID

1.1 Introduction

FLAC3D models the flow of fluid through a permeable solid, such as soil. The flow modeling maybe done by itself, independent of the usual mechanical calculation of FLAC3D, or it may be donein parallel with the mechanical modeling, in order to capture the effects of fluid/solid interaction.One type of fluid/solid interaction is consolidation, in which the slow dissipation of pore pressurecauses displacements to occur in the soil. This type of behavior involves two mechanical effects.First, changes in pore pressure cause changes in effective stress, which affect the response of thesolid (for example, a reduction in effective stress may induce plastic yield). Second, the fluid in azone reacts to mechanical volume changes by a change in pore pressure.

The code handles both fully saturated flow and flow in which a phreatic surface develops. In thiscase, pore pressures are zero above the phreatic surface and the air phase is considered as passive.The logic is applicable to coarse materials when capillary effects can be neglected.

The following characteristics are provided with the fluid-flow capability:

1. Two fluid-transport laws corresponding to isotropic and anisotropic permeability areavailable. The fluid-flow null model is also provided to specify impermeable materialsin the flow domain.

2. Different zones may have different fluid-flow models (isotropic, anisotropic or null) andproperties.

3. Fluid pressure, flux, and leaky and impermeable boundary conditions may be prescribed.

4. Fluid sources (wells) may be inserted into the material as either point sources or volumesources. These sources correspond to either a prescribed inflow or outflow of fluid, andvary with time.

5. Both explicit and implicit fluid-flow solution algorithms are available for fully saturatedsimulations. An explicit method of solution is used for saturated/unsaturated flow.

6. Any of the mechanical and thermal models may be used with the fluid-flow models. Incoupled problems, the compressibility and thermal expansion of the saturated materialare allowed.

7. Coupling between fluid and mechanical calculations due to deformable grains is providedthrough the Biot coefficient, α.

8. Coupling to the thermal conduction calculation is provided through the linear thermalexpansion coefficient, αt , and the undrained thermal coefficient, β.

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1 - 2 Fluid-Mechanical Interaction

9. The thermal-conduction fluid-flow logic is based on a linear theory that assumes constantmaterial properties and neglects convection. Fluid and solid temperatures are locallyequilibrated. Nonlinear behavior can be specified by access to pore pressures and materialproperties via FISH. Thermal coupling to the conduction logic is addressed in Section 1.2.

10. An advection model to take the transport of heat by convection into account is alsoprovided. This includes temperature-dependent fluid density and thermal advection bythe fluid. See Section 1.2.3 in Thermal Analysis for a description of this thermal-fluidcoupling.

Dynamic pore-pressure generation and liquefaction due to cyclic loading can also be modeled withFLAC3D (Section 1.4.4 in Dynamic Analysis contains the documentation on this topic). FLAC3D

does not represent capillary, electrical or chemical forces between particles of a partially saturatedmaterial. However, it is possible to introduce such forces by writing a FISH function that suppliesthe appropriate internal stresses, based on the local saturation, porosity and/or any other relevantvariable. Similarly, the effect of variable fluid stiffness due to dissolved air is not explicitly modeled,but a FISH function may be used to vary the local fluid modulus as a function of pressure, time orany other quantity.

This section is divided into eight major parts:

1. The mathematical model description and the corresponding numerical formulation forfluid flow and coupled fluid flow-mechanical and fluid-flow thermal-conduction processesare described in Sections 1.2 and 1.3.

2. The calculation modes and associated commands for analyses involving fluid flow aredescribed in Section 1.4.

3. Section 1.5 discusses the material properties required for a fluid-flow analysis, and in-cludes the appropriate units for these properties.

4. Section 1.6 talks about initial conditions and provides a description of the differentboundary conditions and fluid sources and sinks that can be applied in a FLAC3D model.

5. The recommended procedures for solving both flow-only and coupled-flow problems areoutlined in Section 1.7. This section also contains several examples that illustrate theapplication of these procedures. We recommend that you work through these examplesbefore attempting your own fluid analysis.

6. Section 1.8 presents several verification problems that demonstrate the accuracy of thefluid-flow logic in FLAC3D.

7. Modeling techniques for specific fluid applications are described in Section 1.9. Sev-eral topics are covered: modeling solid weight, buoyancy forces and seepage forces ina coupled analysis (Section 1.9.1); relation between initialization of pore pressures anddeformation in a coupled analysis (Section 1.9.2); effect of the Biot coefficient (Sec-tion 1.9.3); and modeling a semi-confined aquifer (Section 1.9.4).

8. Finally, a summary of all the commands related to fluid flow is given in Section 1.10.

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The user is strongly encouraged to become familiar with the operation of FLAC3D for simplemechanical problems before attempting to solve problems in which flow and mechanical effectsare both important. Coupled flow and mechanical behavior are often very complicated and requirea good deal of insight to interpret correctly. Before starting a big project, it is very important tospend time experimenting with a small-grid version of the proposed simulation, to try out variousboundary conditions and modeling strategies. The time “wasted” on these experiments will beamply repaid in terms of an overall reduction in staff time, and execution time.

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1.2 FLAC3D Fluid-Thermal-Mechanical Formulation – Mathematical Description

In this section, we consider fluid-mechanical coupling to the heat-conduction logic (see Section 1.2.3in Thermal Analysis for fluid-thermal coupling via the convection logic). Most engineering anal-yses involving simultaneous deformation and fluid/thermal diffusion mechanisms are carried outusing uncoupling techniques (e.g., Section 1.7). In addition to those calculation modes, FLAC3D

provides for the option of coupled fluid-thermal-mechanical analysis (i.e., in which the mechanicalresponse of a porous material can be studied under transient fluid flow and/or thermal condi-tions). Although this section is mainly concerned with the modeling of deformation-fluid diffusionproblems, the general equations governing the fluid-thermal-mechanical response in FLAC3D arepresented here for completeness.

The formulation of coupled deformation-diffusion processes in FLAC3D is done within the frame-work of the quasi-static Biot theory, and can be applied to problems involving single-phase Darcyflow in a porous medium. Various types of fluids, including water and oil, can be represented withthis model.

The variables involved in the description of fluid flow through porous media are pore pressure,saturation and the three components of the specific discharge vector. These variables are relatedthrough the fluid mass-balance equation, Darcy’s law for fluid transport, a constitutive equationspecifying the fluid response to changes in pore pressure, saturation, volumetric strains and tem-perature, and an equation of state relating pore pressure to saturation in the unsaturated range. Porepressure and temperature influences are involved in the mechanical constitutive laws to completethe thermal fluid-flow mechanical coupling.

Assuming the volumetric strain and temperature rates are known, substitution of the mass balanceequation into the fluid constitutive relation, using Darcy’s law, yields a differential equation in termsof pore pressure and saturation that may be solved for particular geometries, properties, boundaryand initial conditions.

1.2.1 Conventions and Definitions

As a notation convention, the symbol ai denotes component i of the vector {a} in a Cartesiansystem of reference axes; Aij is component (i, j) of tensor [A]. Also, f,i is used to represent thepartial derivative of f with respect to xi . (The symbol f denotes either a scalar variable or a vectorcomponent.)

The Einstein summation convention applies only on indices i, j and k, which take the values 1, 2,3 for components that involve spatial dimensions. The indices may take any values when used inmatrix equations.

SI units are used to illustrate parameters and dimensions of variables. See Section 2.9 in the User’sGuide for conversions to other systems of units.

The following reference quantities are useful in the characterization of transient fluid flow.

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Characteristic length:

Lc = volume of f low domain

surf ace area of f low domain(1.1)

Fluid diffusivity:

c = k

1/M + α2/α1(1.2)

where k is mobility coefficient (FLAC3D permeability), M is the Biot modulus, α is the Biotcoefficient, α1 = K + 4/3G, K is the drained bulk modulus, and G is the shear modulus of theporous material.

The Biot coefficient takes into account the grain compressibility for the porous material. If α isequal to unity, the grains are considered to be incompressible and the Biot modulus M is equal toKf /n, where Kf is fluid bulk modulus and n is porosity. The fluid diffusivity becomes

c = k

n/Kf + 1/α1(1.3)

Note that for flow-only calculations (rigid material), the fluid diffusivity is

c = kM (1.4)

See Section 1.5 for additional discussion on the relations between these properties.

Characteristic time:

tc = L2c

c(1.5)

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1.2.2 Governing Differential Equations

The differential equations describing the fluid-thermal-mechanical response of a porous material,for heat transported by conduction in the material domain, are the following:

Transport Law

The fluid transport is described by Darcy’s law:

qi = −kil k(s)[

p − ρf xjgj]

,l(1.6)

where qi is the specific discharge vector, p is fluid pore pressure, k is the tensor of absolute mobilitycoefficients (FLAC3D permeability tensor) of the medium, k(s) is the relative mobility coefficientwhich is a function of fluid saturation s, ρf is the fluid density, and gi , i = 1,3 are the threecomponents of the gravity vector.

For future reference, the quantity φ = (p − ρf xjgj )/(ρf g) (where g is the modulus of the gravityvector) is defined as the head, and ρf gφ as the pressure head.

Heat flow by conduction is described by Fourier’s law of heat transport:

qTi = −kTij T,j (1.7)

where qTi is the heat flux vector, T is temperature and kT is the thermal conductivity tensor. Notethat temperature, flux, convective and adiabatic boundary conditions are considered in the FLAC3D

formulation.

Balance Laws

For small deformations, the fluid mass balance may be expressed as

−qi,i + qv = ∂ζ

∂t(1.8)

where qv is the volumetric fluid source intensity in [1/sec], and ζ is the variation of fluid content orvariation of fluid volume per unit volume of porous material due to diffusive fluid mass transport,as introduced by Biot (1956).

The thermal energy balance is expressed as

−qTi,i + qTv = ∂ζ T

∂t(1.9)

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where ζ T is the heat stored per unit volume of porous material, and qTv is the volumetric heat sourceintensity.

The balance of momentum has the form

σij,j + ρgi = ρdvi

dt(1.10)

where ρ = ρd + nsρw is the bulk density, and ρd and ρw are the densities of the dry matrix andthe fluid, respectively, n is porosity, and s is saturation.

Constitutive Laws

Changes in the variation of fluid content are related to changes in pore pressure, p, saturation, s,mechanical volumetric strains, ε, and temperature, T . The response equation for the pore fluid isformulated as

1

M

∂p

∂t+ n

s

∂s

∂t= 1

s

∂ζ

∂t− α

∂ε

∂t+ β

∂T

∂t(1.11)

where M is Biot modulus [N/m2], n is the porosity, α is Biot coefficient and β is the undrainedthermal coefficient [1/◦C], which takes into account the fluid and grain thermal expansions.

In the FLAC3D formulation, the influence of capillary pressure is neglected and the fluid pressureis taken as zero when saturation is less than one. The relative fluid mobility is related to saturationby a cubic law which is zero for zero saturation, and one for full saturation:

k(s) = s2(3 − 2s) (1.12)

Fluid flow in the unsaturated zone is thus solely driven by gravity. While the influence of gravity isnot required to saturate an initially dry medium, gravity drives the process of desaturation. In thiscase, some level of residual saturation is present because the apparent permeability, k k(s), goes tozero as the saturation approaches zero.

The thermal constitutive law is expressed as

1

MT

∂T

∂t= ∂ζ T

∂t(1.13)

whereMT = 1/(ρCv), ρ is the mass density of the medium, and Cv is the specific heat at constantvolume.

Note that for nearly all solids and liquids, the specific heats at constant pressure and constant volumeare essentially equal. Consequently, Cv and Cρ can be used interchangeably.

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The constitutive response for the porous solid has the form (see notation conventions of Section 1in Theory and Background)

σij + α∂p

∂tδij = H(σij , ξij − ξTij , κ) (1.14)

where the quantity on the left-hand side of the equation is Biot effective stress, σij is the co-rotational stress rate, H is the functional form of the constitutive law, κ is a history parameter, δijis the Kronecker delta, and ξij is the strain rate.

The thermal strain rates are expressed as (see Section 1.2.2.4 in Thermal Analysis)

ξTij = αt∂T

∂tδij (1.15)

where αt is the coefficient of linear thermal expansion.

In particular, the elastic relations which relate effective stresses to strains are (small strain)

σij − σoij + α(P − Po)δij = 2G(εij − εTij )+ (K − 2

3G)(εkk − εTkk) (1.16)

where the superscript o refers to the initial state, εij is the strain, and K and G are the bulk andshear moduli of the drained elastic solid.

Biot and Terzaghi Effective Stress

The expression for Biot effective stress, σij + αpδij , may be derived from compliance principles(see Section 1.9.3); as such, it characterizes the deformability of the solid. On the other hand,Terzaghi effective stress, σij + pδij , measures the stress level sustained by the solid matrix; it isstill being used to detect failure in a plastic material (or potential failure in an elastic solid).

Compatibility Equations

The relation between strain rate and velocity gradient is

ξij = 1

2

[

vi,j + vj,i]

(1.17)

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1.2.3 Fluid Flow Boundary and Initial Conditions in FLAC3D

Initial conditions correspond to known pressure and saturation. Those two values must be consistentwith the FLAC3D formulation: pore pressure must be zero if saturation is less than one, and vice-versa. (By default, in FLAC3D, initial pore pressure is zero and saturation is one.)

Four types of boundary conditions are considered, corresponding to: (1) given pore pressure; (2)given component of the specific discharge normal to the boundary; (3) leaky boundaries; and (4)impermeable boundaries.

In FLAC3D, boundaries are impermeable by default.

A leaky boundary condition has the form

qn = h(p − pe) (1.18)

where qn is the component of the specific discharge normal to the boundary in the direction of theexterior normal, h is the leakage coefficient in [m3/N-sec], p is the pore pressure at the boundarysurface, and pe is the pore pressure in the region to or from which leakage is assumed to occur.

Additional combined boundary conditions, such as uniform pressure for given total influx over aboundary segment, can be imposed using FISH.

Note that saturation cannot be imposed as a boundary condition (this is different from FLAC). Also,to keep the model fully saturated, the fluid tension may be set to a large negative number.

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1.3 Numerical Formulation

1.3.1 Introduction

Substitution of the fluid mass balance Eq. (1.8) in the constitutive equation for the pore fluidEq. (1.11) yields the expression for the fluid continuity equation:

1

M

∂p

∂t+ n

s

∂s

∂t= 1

s(−qi,i + qv)− α

∂ε

∂t+ β

∂T

∂t(1.19)

In the FLAC3D numerical approach, the flow domain is discretized into brick-shaped zones definedby eight nodes. (The same discretization is used for mechanical and thermal calculations, whenapplicable.) Both pore pressure and saturation are assumed to be nodal variables. Internally, eachzone is subdivided into tetrahedra (2 overlays are used by default), in which the pore pressure andthe saturation are assumed to vary linearly.

The numerical scheme relies on a finite difference nodal formulation of the fluid continuity equation.The formulation can be paralleled to the mechanical constant stress formulation (presented inSection 1.1.2.2 in Theory and Background) that leads to the nodal form of Newton’s law. It isobtained by substituting the pore pressure, specific discharge vector and pore-pressure gradient forvelocity vector, stress tensor and strain-rate tensors, respectively. The resulting system of ordinarydifferential equations is solved using an explicit mode of discretization in time (default mode). Animplicit fluid flow calculation scheme is also available. However, it is only applicable to saturatedflow simulations (flow in which the unit saturation is not allowed to change).

The principal results are summarized below. The formulation, as it applies to saturated flow, isdescribed first in Section 1.3.2. This formulation is then extended to apply to saturated/unsaturatedflow simulations in Section 1.3.3. For now we note that the saturated/unsaturated algorithm is basedon variable substitution and upstream weighting techniques and conserves fluid flow.

Starting from a state of mechanical equilibrium, a coupled hydromechanical static simulation inFLAC3D involves a series of steps. Each step includes one or more flow steps (flow loop) followedby enough mechanical steps (mechanical loop) to maintain quasi-static equilibrium.

The increment of pore pressure due to fluid flow is evaluated in the flow loop; the contribution fromvolumetric strain is evaluated in the mechanical loop as a zone value which is then distributed tothe nodes.

For the effective stress calculation, the total stress increment due to pore-pressure change arisingfrom mechanical volume strain is evaluated in the mechanical loop, and that arising from fluid flowis evaluated in the flow loop.

Note that in FLAC3D, all material models are expressed in terms of Terzaghi effective stresses(i.e., effective stresses are used to detect failure in plastic materials). In this context, the pore-pressure field may originate from different sources: a fluid flow analysis; a coupled fluid/mechanicalsimulation; or an initialization with the INI pp or WATER table command.

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1.3.2 Saturated Fluid Flow

In this section, as a first step towards the derivation of the more general FLAC3D fluid flow algorithm,we consider flow in a medium which remains fully saturated at all times.

1.3.2.1 Finite-Difference Approximation to Space Derivatives

By convention, the tetrahedron nodes are referred to locally by a number from 1 to 4, face n isopposite node n, and the superscript (f ) relates to the value of the associated variable on face f .

A linear pore-pressure variation and constant fluid density are assumed within a tetrahedron. Thepressure head gradient, expressed in terms of nodal values of the pore pressure by application ofthe Gauss divergence theorem, may be written as

(p − ρf xigi),j = − 1

3V

4∑

l=1

(pl − ρf xli gi) n

(l)j S(l) (1.20)

where n(l) is the exterior unit vector normal to face l, S is the face surface area, and V is thetetrahedron volume.

For numerical accuracy, the quantity xi −x1i , where x1

i corresponds to the coordinates of one of thetetrahedron’s corners, may be substituted for xi in the expression of the pressure head in Eq. (1.20)without affecting the value of the pressure head gradient. Accordingly, Eq. (1.20) takes the form

(p − ρf xigi),j = − 1

3V

4∑

l=1

p∗l n(l)j S(l) (1.21)

where the nodal quantity p∗l is defined as

p∗l = pl − ρf (xli − x1

i )gi (1.22)

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1.3.2.2 Nodal Formulation of the Mass Balance Equation

For fully saturated flow (s is constant and equal to one), the fluid continuity Eq. (1.19) may beexpressed as

qi,i + b∗ = 0 (1.23)

where

b∗ = S

M

∂p

∂t− q∗

v (1.24)

is the equivalent of the instantaneous “body forces,” ρbi , used in the mechanical node formulation,and

q∗v = qv − α

∂ε

∂t+ β

∂T

∂t(1.25)

First consider a single tetrahedron. Using this analogy, the nodal discharge, Qne [m3/s], n = 1,4,

equivalent to the tetrahedron-specific discharge and the volumetric source intensity, b∗, may beexpressed as (see Section 1.1.2.2 in Theory and Background)

Qne = Qn

t − q∗vV

4+mn

dp

dt

n

(1.26)

where

Qnt = qin

(n)i S(n)

3(1.27)

and

mn = V

4Mn(1.28)

In principle, the nodal form of the mass balance equation is established by requiring that, at eachglobal node, the sum of equivalent nodal discharges (−Qn

e ) from tetrahedra meeting at the node(averaged over two overlays) and nodal contribution (Qn

w) of applied boundary fluxes and sourcesbe zero.

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The components of the tetrahedron-specific discharge vector in Eq. (1.27) are related to the pressure-head gradient by means of the transport law (see Eq. (1.6)). In turn, the components of the pressure-head gradient can be expressed in terms of the tetrahedron nodal pore pressures, using Eq. (1.21).

In order to save computer time, local matrices are assembled. A zone formulation is adoptedwhereby the sum Qn

z of contributions Eq. (1.27) from all tetrahedra belonging to the zone andmeeting at a node, n, is formed and averaged over two overlays. Local zone matrices [M] thatrelate the eight nodal values {Qz} to the eight nodal pressure heads {p∗} are assembled. Becausethese matrices are symmetrical, 36 components are computed; these are saved at the beginning ofthe computation and updated every ten steps in large-strain mode. By definition of zone matrices,we have

Qnz = Mnjp

∗j (1.29)

where {p∗} is the local vector of nodal pressure heads for the zone.

In turn, global nodal values QnT are obtained by superposition of zone contributions. Taking some

liberty with the notation, we write for each global node n,

QnT = Cnjp

∗j (1.30)

where [C] is the global matrix and {p∗} is, in this context, the global vector of nodal pressure heads.

Returning to our previous consideration, we write

−∑

Qne +

∑

Qnw = 0 (1.31)

where, for simplicity of notation, the∑

sign is used to represent summation of the contributions atglobal node n of all zones meeting at that node. (A zone contribution consists of contributions oftetrahedra involved in the zone, averaged over two overlays.) Using Eq. (1.26) forQn

e and Eq. (1.25)for q∗

v in Eq. (1.31), we obtain, after some manipulation,

dpn

dt= − 1

∑

mn

[

QnT +

∑

Qnapp +

∑

Qnthm

]

(1.32)

where QnT is a function of the nodal pore pressures defined in Eq. (1.30) together with Eq. (1.22),

∑

Qnapp is the known contribution of applied volume sources, boundary fluxes and point sources,

having the form

∑

Qnapp = −

∑

[

qvV

4+Qw

]n

(1.33)

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and∑

Qnthm is the thermal-mechanical nodal discharge contribution defined as

∑

Qnthm = d

dt

[

∑

(

αεV

4

)n

−∑

(

βTV

4

)n]

(1.34)

Eq. (1.32) is the nodal form of the mass-balance equation at node n; the right-hand side termQnT + ∑

Qnapp + ∑

Qnthm is referred to as the out-of-balance discharge. This term is made up

of two contributions: the out-of-balance flow discharge, QnT + ∑

Qnapp; and the out-of-balance

thermal-mechanical discharge,∑

Qnthm. When the fluid is at rest (the fluid-flow calculation is

turned off), the out-of-balance flow discharge vanishes and pore-pressure changes are caused bymechanical and/or thermal deformations only. For flow calculation only (no thermal-mechanicalcoupling), the out-of-balance thermal-mechanical discharge is equal to zero.

In FLAC3D, the Biot modulus is a nodal property, and we may write, using Eq. (1.28),

1∑

mn= Mn

V n(1.35)

where

V n =∑

(

V

4

)n

(1.36)

After substitution of Eq. (1.34) in Eq. (1.32), we write, rearranging terms,

d

dt

[

pn − pnv] = −M

n

V n

[

QnT +

∑

Qnapp

]

(1.37)

where

pnv = −Mn

V n

[

∑

(

αεV

4

)n

−∑

(

βTV

4

)n]

(1.38)

An expression such as Eq. (1.37) holds at each global node involved in the discretization. Together,these expressions form a system of ordinary differential equations that is solved in FLAC3D, forgiven dpnv/dt , using either explicit or implicit (saturated flow only) finite-difference schemes. Thedomain of application of each scheme is discussed below.

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1.3.2.3 Explicit Finite-Difference Formulation

In the explicit formulation, the value of the quantityp−pv at a node is assumed to vary linearly overa time interval t . In principle, the derivative in the left-hand side of Eq. (1.37) is expressed usingforward finite differences, and the out-of-balance flow-discharge is evaluated at time t . Startingwith an initial pore-pressure field, nodal pore pressures at incremental time values are updated,provided the pore pressure is not fixed, using the expression

pn<t+ t> = pn<t> + pnv<t> + pn<t> (1.39)

where

pn<t> = χn[

QnT<t> +

∑

Qnapp<t>

]

(1.40)

χn = −Mn

V n t (1.41)

and

pnv<t> = −Mn

V n

[

∑

(

α εV

4

)n

−∑

(

β TV

4

)n]

<t>

(1.42)

Numerical stability of the explicit scheme can only be achieved if the timestep remains below alimiting value.

Stability Criterion

To derive the stability criterion for the flow calculation, we consider the situation in which a node,n, in an assembly of zones is given a pore-pressure perturbation, p0, from a zero initial state. UsingEq. (1.30), we obtain

QnT = Cnn p0 (1.43)

If node n belongs to a leaky boundary, we have

∑

Qnapp = Dnn p0 (1.44)

where Dnn is used to represent the pressure coefficient in the global leakage term at node n.

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After one fluid-flow timestep, the new pore pressure at node n is (see Eqs. (1.39) through (1.50))

pn< t> = p0

[

1 − Mn

V n(Cnn +Dnn) t

]

(1.45)

To prevent alternating signs of pore pressures as the computation is repeated for successive t , thecoefficient of p0 in the preceding relation must be positive. Such a requirement implies that

t <V n

Mn

1

Cnn +Dnn(1.46)

The value of the poro-mechanical timestep used in FLAC3D is the minimum nodal value of theright-hand side in Eq. (1.46), multiplied by the safety factor of 0.8.

To assess the influence of the parameters involved, it is useful to keep in mind the followingrepresentation of the critical timestep. If Lc is the smallest tetrahedron characteristic length, wemay write an expression of the form

tcr = 1

a

[

c

L2c

+ hM

Lc

]−1

(1.47)

where tcr is the critical timestep, c is the fluid diffusivity, and a is a constant, larger than unity,that depends on the medium geometrical discretization (e.g., a = 6 for a regular discretization incubes; see Karlekar and Desmond (1982) for a thermal analogy).

The critical timestep in Eq. (1.47) corresponds to a measure of the characteristic time needed forthe diffusion “front” to propagate throughout the tetrahedron. To estimate the time needed for thefront to propagate throughout a particular domain, a similar expression can be used, provided Lc isinterpreted as the characteristic length of the domain under consideration. Consider the case whereno leakage occurs and the properties are homogeneous. By taking the ratio of characteristic timesfor the domain and the tetrahedron, we see that the number of steps needed to model the propagationof the diffusion process throughout that domain is proportional to the ratio of square characteristiclengths for the domain and the tetrahedron. That number may be so large that the use of the explicitmethod alone could become prohibitive. However, in many groundwater problems, the advantageof this first-order method is that the calculated timestep is usually small enough to follow nodalpore-pressure fluctuations accurately.

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1.3.2.4 Implicit Finite-Difference Formulation

The requirement that t should be restricted in size to ensure stability sometimes results in anextremely small timestep (of the order of a fraction of a second) especially when transient flow inlayers of contrasting properties is involved. The implicit formulation eliminates this restriction, butit involves solving simultaneous equations at each timestep and, besides, it only applies to saturatedfluid flow simulations.

The implicit formulation in FLAC3D uses the Crank-Nicolson method, in which the pore pressurep − pv at a node is assumed to vary quadratically over the time interval t . In this method,the derivative d(p − pv)/dt in Eq. (1.37) is expressed using a central difference formulationcorresponding to a half timestep, while the out-of-balance flow discharge is evaluated by takingaverage values at t and t + t . In this formulation we have

pn<t+ t> = pn<t> + pnv<t> + pn<t> (1.48)

and

pn<t> = χn[

1

2

(

QnT<t+ t> +Qn

T<t>

)+∑

Qnapp

]

(1.49)

where

χn = −Mn t

V n(1.50)

pnv<t> is given by Eq. (1.42), and

∑

Qnapp = 1

2

(∑

Qnapp<t+ t> +

∑

Qnapp<t>

)

(1.51)

From Eq. (1.30), we may write

QnT<t> = Cnjp

∗j<t> (1.52)

and

QnT<t+ t> = Cnjp

∗j<t+ t> (1.53)

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For no variation of the gravity term during the time interval t , Eq. (1.48) can be written usingEq. (1.22).

p∗n<t+ t> = p∗n

<t> + pnv<t> + pn<t> (1.54)

Using this last expression and Eq. (1.52), Eq. (1.53) becomes

QnT<t+ t> = Qn

T<t> + Cnj pj<t> + Cnj p

jv<t> (1.55)

After substitution of Eq. (1.55) into Eq. (1.49), we obtain, using Eq. (1.52),

pn<t> = χn[

Cnj (p∗j<t> + 1

2 p

jv<t>)+ 1

2Cnj p

j<t> +

∑

Qnapp

]

(1.56)

Finally, regrouping terms:

[

δnj − χn

2Cnj

]

pj<t> = χn

[

Cnj (p∗j<t> + 1

2 p

jv<t>)+

∑

Qnapp

]

(1.57)

For simplicity of notation, we define the known matrix [A] and vector [b<t>] as

Anj = δnj − χn

2Cnj (1.58)

and

bn<t> = χn[

Cnj (p< t >∗j + 1

2 p

jv<t>)+

∑

Qnapp

]

(1.59)

With this convention, we may write Eq. (1.57) in the form

Anj pj<t> = bn<t> (1.60)

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One equation holds for each of the nodes involved in the grid, and the resulting system is solvedin FLAC3D using Jacobi’s iterative method. In this approach, pore-pressure increments at iterationr + 1 and node n are calculated using the recurrence relation

pn<r+1><t> = pn<r><t> + 1

Ann

[

−Anj pj<r><t> + bn<t>

]

(1.61)

where Einstein’s summation convention applies to index j only. Using the Eq. (1.58) of [A],Eq. (1.61) takes the form

pn<r+1><t> = pn<r><t> + 1

1 − χn

2 Cnn

[

χn

2Cnj p

j<r><t> − pn<r><t> + bn<t>

]

(1.62)

The initial approximation is chosen such that

pn<0><t> = 0 (1.63)

(Note that, in FLAC3D, pressure-dependent boundary conditions (contained in bn) are updated inthe implicit iterative procedure.)

Convergence Criterion

In FLAC3D, a minimum of 3 and a maximum of 500 iterations are considered, and the criterion fordetection of convergence has the form

maxn

∣

∣ pn<r+1><t> − pn<r><t>

∣

∣ < 10−2(

maxn

∣

∣ pn<r><t>

∣

∣

)

(1.64)

The magnitude of the timestep must be selected in relation to both convergence and accuracy ofthe implicit scheme. Although the Crank-Nicolson method is stable for all positive values of t(for no leakage), the convergence of Jacobi’s method is not unconditionally guaranteed unless thematrix [A] is strictly diagonally dominant:

n∑

j=1j �=k

∣

∣

∣

∣

Akj

∣

∣

∣

∣

<

∣

∣

∣

∣

Akk

∣

∣

∣

∣

(1.65)

for 1 ≤ k ≤ n (see Dahlquist and Bjorck 1974). According to the definitions of Anj (Eq. (1.58))and χn (Eq. (1.50)), this sufficient condition is always fulfilled for sufficiently small values of t .If convergence of Jacobi’s method is not achieved, an error message is issued. It is then necessary

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to either reduce the magnitude of the implicit timestep, or use the explicit method. (The explicittimestep may be used as a lower bound.)

Also, although the implicit method is second-order accurate, some insight may be needed in orderto select the appropriate timestep. Indeed, its value must remain small compared to the wavelengthof any nodal pore-pressure fluctuation. Typically, the explicit method is used earlier in the run orin its perturbed stages, while the implicit method is preferred for the remainder of the simulation.(Alternatively, the implicit method could be used with the explicit timestep value for extra accuracy.)

It must be remembered that the implicit approach should not be selected for simulations in whichchanges of saturation are to be expected. In addition, computation time and computer memory aretwo factors that must be taken into consideration when selecting the implicit approach in FLAC3D.In the implicit method, a set of equations requiring a minimum of three iterations must be solved ateach timestep. The amount of calculation required for one iterative step is approximately equal tothat needed for one timestep in the explicit scheme. Also, intermediate values must be stored in theiterative procedure, requiring extra memory to be allocated in comparison with the explicit scheme.Those inconveniences, however, can be more than offset by the much larger timestep generallypermitted by the implicit method, or by the gain in accuracy.

1.3.3 Saturated/Unsaturated Flow

For saturated fluid flow, the nodal volumetric flow rates in a zone {Qz} are related to the nodal porepressures {p} by Eq. (1.29), which may be expressed in matrix notation as

{Qz} = [M]{p − ρf xigi} (1.66)

where the matrix [M] is a known function of the zone geometry and the saturated value of themobility coefficient. This relation was derived by application of the Gauss divergence theoremfor pore-pressure gradient in the tetrahedron, taking into account energy considerations, and usingsuperposition. It is extended to the case of unsaturated flow in coarse soils (constant air pressure,no capillary pressure) by application of the following modifications:

(a) The gravity term ρf xigi is multiplied by the average zone saturation, s, to account forpartial zone filling.

(b) The nodal flow rates are multiplied by the relative mobility, k (see Eq. (1.6)), which isa function of the average saturation at the inflow nodes for the zone, sin. This upstreamweighting technique is applied to prevent unaccounted flow in the filling process of aninitially dry medium (numerical dispersion caused by the nodal definition of saturation).The relative permeability function has the same form as Eq. (1.12):

k(sin) = s2in(3 − 2sin) (1.67)

(c) Nodal inflow rates are scaled according to local saturation.

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For saturated/unsaturated flow, the explicit finite difference nodal formulation of the fluid continuityEq. (1.19) takes the form (see Eqs. (1.39) to (1.42))

p

nM+ s

s= − 1

snV

[

(

QT +∑

Qapp

)

t + s

(

∑

α εVT

4−∑

β TVT

4

)]

(1.68)

where the subscript < t > and superscript n have been omitted for simplicity of notation, and VTstands for tetrahedron volume. At a saturated node, s = 1 and Eq. (1.68) becomes

p = −MV

[

(

QT +∑

Qapp

)

t +(

∑

α εVT

4−∑

β TVT

4

)]

(1.69)

At a partially saturated node, p = 0 and Eq. (1.68) may be expressed as

s = − 1

nV

[

(

QT +∑

Qapp

)

t + s

(

∑

α εVT

4−∑

β TVT

4

)]

(1.70)

The approach leads to a technique of variable substitution whereby the saturation variable is updatedusing Eq. (1.70) at an unsaturated node (where the pore pressure is equal to zero), and Eq. (1.69)is used to increment the pore-pressure variable at a saturated node (where saturation is constant).Those relations are applied in such a way that the fluid volume balance is preserved (i.e., Eq. (1.68)is satisfied).

The explicit fluid timestep in FLAC3D is selected on the basis of a stability criterion derived using thesaturated fluid diffusivity c = kM (k is the largest principal value of saturated mobility coefficienttensor, M is Biot modulus). (See Eqs. (1.46) and (1.47).) The time scale associated with fluidstorage is usually much smaller than the characteristic time scale for phreatic storage. While anestimate of the first one (two-dimensional flow) can be calculated using t ∝ L2/c, where L is tobe interpreted as average flow path length, a rough estimate of the second may be obtained fromthe relation t ∝ L2/(kp/n), where L is the average height of the medium associated with phreaticstorage, and p is the related mean pore-pressure change.

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1.3.4 Mechanical Timestep and Numerical Stability

The presence of the fluid will increase the apparent mechanical bulk modulus of the medium which,in turn, affects the magnitude of the nodal mass in the density scaling scheme used for numericalstability in FLAC3D. An upper bound for this apparent modulus may be found by consideringundrained, saturated isothermal poro-mechanical conditions for which ∂ζ

∂t= 0, ∂s

∂t= 0 and ∂T

∂t= 0.

The incremental expression of the fluid constitutive law Eq. (1.11) then takes the form

p = −αM ε (1.71)

Adopting the incremental form of the elastic law Eq. (1.16) to describe the volumetric part of thestress-strain constitutive relation in a time interval, we can write

1

3 σii + α p = K ε (1.72)

Substitution of Eq. (1.71) in Eq. (1.72) gives

1

3 σii = (K + α2M) ε (1.73)

This apparent tangent bulk modulus,K+α2M , is used in place ofK when calculating nodal inertialmass (see Eq. (1.64) in Theory and Background).

Also note that in FLAC3D, the dry density, ρd , of the medium is considered in the input. Thesaturated density, ρ = ρd + nsρf , is calculated internally using the input value for fluid density,ρf , porosity, n, and saturation, s. Body forces are then adjusted accordingly in Newton’s equationsof motion.

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1.3.5 Total Stress Correction

In the FLAC3D formulation, nodal pore pressures are calculated first. The zone pore pressures arethen derived by volume averaging of the tetrahedra values (obtained as arithmetic averages of nodalvalues).

The total stresses are then adjusted, consistent with the definition of Biot effective stress, by addingan increment σij to account for the contribution from pore pressure changes (see Eq. (1.14)). Thestress increment σij is expressed as

σij = σfij + σthij (1.74)

where

σfij = −α pn<t>δij (1.75)

is the contribution from flow, and

σthij = αM(α ε − β T )<t>δij (1.76)

is the contribution from thermal-mechanical coupling. Also, in those expressions, the overlinesymbol stands for zone average in the sense mentioned above.

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1.4 Calculation Modes for Fluid-Mechanical Interaction

The calculation mode and commands required for a pore-pressure analysis depend on whether thegrid has been configured for fluid flow (i.e., whether the CONFIG fluid command has been specified).Sections 1.4.1 and 1.4.2 reflect the two possibilities. For convenience, all commands describedbelow are summarized in Section 1.10, at the end of this section.

1.4.1 Grid Not Configured for Fluid Flow

If the command CONFIG fluid has not been given, then a fluid-flow analysis cannot be performed,but it is still possible to assign pore pressure at gridpoints. In this calculation mode, pore pressuresdo not change, but failure, which is controlled by the effective-stress state, may be induced whenplastic constitutive models are used.

A pore-pressure distribution can be specified at gridpoints, with the INITIAL pp command with agradient, or with the WATER table command.

If the WATER table command is used, a hydrostatic pore-pressure distribution is calculated auto-matically by the code, below the given water table level. In this case, the fluid density (WATERdensity) and gravity (SET gravity) must also be specified. The fluid density value and water tablelocation can be printed with the LIST water command, and the water table can be plotted with thePLOT water command, if the water table is defined with the face keyword.

In both cases, zone pore pressures are calculated by averaging from the gridpoint values, and usedto derive effective stresses for use in the constitutive models. In this calculation mode, the fluidpresence is not automatically accounted for in the calculation of body forces: wet and dry mediumdensities must be assigned by the user, below and above the water level, accordingly. The commandsLIST gp pp and LIST zone pp print gridpoint and zone pore pressures, respectively. PLOT contourgpp plots contours of gridpoint pore pressures.

A simple example application of this calculation mode is given in Section 3.6.3 in the User’s Guide.

1.4.2 Grid Configured for Fluid Flow

If the command CONFIG fluid is given, a transient fluid-flow analysis can be performed, and change inpore pressures, as well as change in the phreatic surface, can occur. Pore pressures are calculated atgridpoints, and zone values are derived using averaging. Both effective-stress (static pore-pressuredistribution) and undrained calculations can be carried out in CONFIG fluid mode. In addition, afully coupled analysis can be performed, in which changes in pore pressure generate deformation,and volumetric strain causes the pore pressure to evolve.

If the grid is configured for fluid flow, dry densities must be assigned by the user (both below andabove the water level), because FLAC3D takes the fluid influence into account in the calculation ofbody forces.

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A fluid-flow model must be assigned to zones when running in CONFIG fluid mode. Isotropic flow isprescribed with the MODEL fluid fl isotropic command; anisotropic flow with the command MODELfluid fl anisotropic. An impermeable material is assigned to zones with the MODEL fluid fl nullcommand. Note that zones that are made null mechanically are not automatically made null forfluid flow.

Fluid properties are assigned to either zones or gridpoints. Zone fluid properties include isotropicpermeability, porosity, Biot coefficient, undrained thermal coefficient and fluid density. Zone fluidproperties are assigned with the PROPERTY command, with the exception of fluid density, whichis assigned with the INITIAL command. (Note that fluid density can also be specified globally withthe WATER command.)

For isotropic flow, permeability is specified using the perm property keyword. For anisotropic flow,the three principal values of permeability are specified using the property keywords k1, k2 and k3,and the orientation is defined using the keywords fdip, fdd and frot. The principal directions ofpermeability correspond to k1, k2, k3, and form a right-handed system. The properties fdip and fddare the dip angle and dip direction angle of the plane in which k1 and k2 are defined. The dip angleis measured from the global xy-plane, positive down (in negative z-direction). The dip directionangle is the angle between the positive y-axis and the projection of the dip-direction vector onthe xy-plane (positive clockwise from y-axis). The property frot is the rotation angle between thek1-axis and the dip vector in the k1-k2-plane (positive clockwise from dip-direction vector). SeeFigure 1.1.

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Figure 1.1 Definitions for principal directions of permeability tensor

Biot coefficient is assigned using the biot c keyword; porosity is assigned using the poros keyword.If not specified, Biot coefficient = 1 and porosity = 0.5, by default.

Gridpoint fluid properties are assigned with the INITIAL command. These properties include fluidbulk modulus, Biot modulus, fluid tension limit and saturation. Each of the gridpoint propertiescan also be given a spatial variation.

Table 1.1 summarizes the ways the various properties can be specified. The fluid properties aredescribed further in Section 1.5.

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Table 1.1 Property specification methods

property keyword specified command

Permeability (isotropic) perm in zones PROPERTYPerm., princ. val. (aniso) k1 in zones PROPERTYPerm., princ. val. (aniso) k2 in zones PROPERTYPerm., princ. val. (aniso) k3 in zones PROPERTYPerm., princ. dir. (aniso) fdip in zones PROPERTYPerm., princ. dir. (aniso) fdd in zones PROPERTYPerm., princ. dir. (aniso) frot in zones PROPERTYPorosity poros in zones PROPERTYBiot coefficient biot c in zones PROPERTYUndrained thermal coeff. u thc in zones PROPERTYFluid modulus fmod at gridpoints INITIALBiot modulus biot mod at gridpoints INITIALSaturation sat at gridpoints INITIALFluid tension limit ftens at gridpoints INITIALFluid density fdens at zones INITIAL

dens globally WATER

It is important to note that fluid compressibility is defined in one of two ways in the CONFIG fluidmode: 1) Biot coefficient and Biot modulus are specified; or 2) fluid bulk modulus and porosity arespecified. The first case accounts for the compressibility of the solid grains (Biot coefficient is setto 1 for incompressible grains). In the second case, solid grains are assumed to be incompressible(see Section 1.5.3.1).

The zone properties can be printed with the LIST zone property command, and the gridpoint prop-erties can be printed with the LIST gp command. Fluid density, along with the location of the watertable, if specified, can be printed with the LIST water command. The fluid-flow properties can beplotted as a contour plot with the PLOT bcontour property command, or as a block plot with thePLOT block property command. For anisotropic flow, the global components of the permeabilitytensor are available for plotting and printing, using the zone property keywords kxx, kyy, kzz, kxy,kxz and kzz (please note that these global components cannot be initialized directly).

An initial gridpoint pore-pressure distribution is assigned the same way for the CONFIG fluid modeas for the non-CONFIG fluid mode (i.e., either with the INITIAL pp command or the WATER tablecommand). Pore pressures can be fixed (and freed) at selected gridpoints with the FIX pp (andFREE pp) command. Fluid sources and sinks can be applied with the APPLY command. Section 1.6describes the various fluid-flow boundary and initial conditions.

The fluid-flow solution is controlled by the SET fluid and SOLVE commands. Several keywords areavailable to help the solution process. For example, SET fluid on or off turns on or off the fluidflow calculation mode. The application of these commands and keywords depends on the level ofcoupling required in the fluid flow analysis. Section 1.7 describes the various coupling levels and

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the recommended solution procedure. Example applications, ranging from flow-only to coupledmechanical-flow calculations, are also described in this section.

Results from a fluid-flow analysis are provided in several forms. The commands LIST gp pp andLIST zone pp print gridpoint and zone pore pressures, respectively. Histories of gridpoint andzone pore pressures can be monitored with the HISTORY gp pp and HISTORY zone pp commands,respectively. And for a transient calculation, the pore pressure can be plotted versus real time bymonitoring the flow time with the HISTORY fltime command. PLOT contour gpp plots contours ofgridpoint pore pressures. PLOT contour saturation plots contours of saturation. The PLOT fluidcommand plots specific discharge vectors. General information on the CONFIG fluid calculationmode is printed with the LIST fluid command. Several fluid-flow variables can be accessed throughFISH. These are listed in Section 1.10.2. There is one grid-related variable, gp flow, that can onlybe accessed through a FISH function; this corresponds to the net inflow or outflow at a gridpoint.The summation of such flows along a boundary where the pore pressure is fixed is useful becauseit can provide a value for the total outflow or inflow for a system.

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1.5 Properties and Units for Fluid-Flow Analysis

The properties that relate to fluid flow in FLAC3D are the permeability coefficient, k, the fluidmass density, ρf , and either the Biot coefficient, α, and Biot modulus, M , for flow through amaterial with compressible grains, or the fluid bulk modulus, Kf , and porosity, n, for flow througha material with incompressible grains only. The thermal-coupling parameters are the coefficient ofdrained linear thermal expansion, αt (thermal-mechanical), and the undrained thermal coefficient,β (poro-thermal). These properties are examined below.

All thermal-poro-mechanical quantities must be given in a consistent set of units. No conversionsare performed by the program.

1.5.1 Permeability Coefficient

The isotropic permeability coefficient, k (e.g., m2/(Pa/sec) in SI units), used in FLAC3D is alsoreferred to in the literature as the mobility coefficient. It is the coefficient of the pressure term inDarcy’s law and is related to the hydraulic conductivity, kh (e.g., m/s), by the expression

k = kh

ρf g(1.77)

where g is the gravitational acceleration.

The intrinsic permeability, κ (e.g., in m2), is related to k as follows:

k = κ

μ(1.78)

where μ is the dynamic viscosity (e.g., units of N-sec/m2).

Eq. (1.77) or Eq. (1.78) may be used to derive k (required by FLAC3D ) from either kh, in velocityunits, or κ , in [length]2 units (remember that kmust end up with units [L3T/M] – e.g., in SI units thiswould be m3 sec/kg (or m2/Pa-sec))). Using the values μ = 1.011 ×10−3 kg/(m-sec) for water at20◦ C, ρwg = 9.81 ×103 Pa/m and 1 Darcy = 10−8 cm2, the following conversions may be derivedto calculate k in SI units for water in FLAC3D :

k (in SI units) ≡ κ (in cm2) × 9.9 × 10−2

k (in SI units) ≡ kh (in cm/sec) × 1.02 × 10−6

k (in SI units) ≡ permeability in millidarcies × 9.8 × 10−13

If there is a variation of permeability across the grid, the timestep will be controlled by the largestpermeability (see Eq. (1.47)). For problems in which steady state (but not transient behavior) isrequired, it may be beneficial to limit the variations in permeability to improve convergence speed.

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For example, there will probably be little difference in the final state between systems where thereis a 20:1 variation in permeability, compared to a 200:1 variation, when the flow is moving entirelyfrom one permeability region to the other.

The permeability coefficient (scalar or tensor) is a zone property specified using the PROPERTYcommand (see Section 1.4.2).

1.5.2 Mass Density

Three different mass densities may be given as input to FLAC3D in different circumstances: the drydensity of the solid matrix, ρd ; the saturated density of the solid matrix, ρs ; and the density of thefluid, ρf . Note that densities are only required if gravitational loading is specified.

If FLAC3D is configured for fluid flow (CONFIG fluid), then the dry density of the solid materialmust be used. FLAC3D will compute the saturated density of each element using the known densityof the fluid, the porosity, n, and the saturation, s: ρs = ρd + nsρf .

The only case when the saturated density is given as input is for an effective stress calculation (staticpore-pressure distribution) not carried out in CONFIG fluid mode. The WATER table command (orINITIAL pp command) specifies the location of the water table. The dry density is specified for zonesabove the water table, and the saturated density for zones below.

The solid density (dry or saturated) is given using the INITIAL density command. The fluid densitycan be imposed globally using the WATER density command, or it can be allowed to vary withposition by using the INITIAL fdensity command. However, this capability of assigning spatialvariation of fluid density must only be used with extreme caution because fluid mass density isnot advected in FLAC3D. All densities are zone variables in FLAC3D, and are mass densities (e.g.,kg/m3 in SI units).

1.5.3 Fluid Moduli

1.5.3.1 Biot Coefficient and Biot Modulus

The Biot coefficient,α, is defined as the ratio of the fluid volume gained (or lost) in a material elementto the volume change of that element when the pore pressure is changed. It can be determined in thesame drained test as that used to determine the drained bulk modulus,K , of the material. Its range ofvariation is between 3n

2+n and 1, where n is the porosity. In the particular case of an incompressiblesolid constituent, α = 1. This value is the default value adopted by FLAC3D.

For an ideal porous material, the Biot coefficient is related to the bulk modulus of the solid componentKs :

α = 1 − K

Ks(1.79)

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The Biot modulus, M , is defined as

M = Ku −K

α2(1.80)

where Ku is the undrained bulk modulus of the material.

For an ideal porous material, the Biot modulus is related to the fluid bulk modulus, Kf :

M = Kf

n+ (α − n)(1 − α)Kf /K(1.81)

where n is the porosity. Thus, for an incompressible solid constituent (α = 1),

M = Kf /n (1.82)

The calculation mode for compressible grains is turned on with the SET fluid biot on command. TheBiot coefficient is a zone property specified using the PROPERTY command. The Biot modulus isa gridpoint variable specified using the INITIAL command.

1.5.3.2 Fluid Bulk Modulus

In analyses where the grain compressibility can be neglected, the user has the choice to either usethe default value of Biot coefficient (i.e., α = 1) and assign a value equal toKf /n to Biot modulus,or give the fluid bulk modulus Kf as input.

The bulk modulus, Kf , is defined as

Kf = P

Vf /Vf(1.83)

where P is the change in pressure for a volumetric strain of Vf /Vf .

When the fluid modulus Kf is given as an input, the Biot modulus is computed internally usingEq. (1.82) for incompressible grains. In this calculation, the porosity (a zone property) is evaluatedat the nodes using nodal volume averaging. The Biot coefficient is then set to 1 throughout the flowdomain, irrespective of any value given for that property. The fluid calculation for incompressiblegrains is the default calculation in FLAC3D (i.e., SET fluid biot off). The fluid modulus is a gridpointvariable specified using the INITIAL command.

The “compressibility” of the fluid Cf is the reciprocal of Kf (i.e., Cf = 1/Kf ). For example,for pure water at room temperature, Kf = 2 × 109 Pa, in SI units. In real soils, pore water maycontain some dissolved air or air bubbles, which substantially reduce its apparent bulk modulus.

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For example, Chaney (1978) reports that the fluid modulus can be reduced by an order of magnitudefor an air/water mixture at 99% saturation in compacted sand. For groundwater problems, the bulkmodulus of water may be different in different parts of the grid, to account for the varying amountsof air present. A FISH function may be used to change local moduli according to some law (e.g.,the modulus may be made proportional to pressure to represent a gas), but the user should be carefulnot to make nonphysical assumptions.

1.5.3.3 Fluid Moduli and Convergence

If steady-state, fully saturated flow is required, the modulus M (or Kf ) is unimportant to thenumerical convergence process, because the response time of the system and the timestep are bothinversely proportional to M (or Kf ): the same number of steps are necessary, independent ofM (or Kf ). For systems containing a phreatic surface, however, a low bulk modulus will speedconvergence to steady state, because the calculation for saturation change involves t , not theproduct M t (or Kf t) (see Eqs. (1.69) and (1.70)). Systems in which solid/fluid interactionis important are more complicated to assess (some guidelines may be found in Section 1.7.1).However, a high value ofM (orKf ) compared to the mechanicalK will lead to slowly convergingsolutions. In any case, from a numerical point of view, it is not necessary to use values ofM (orKf )that are larger than 20 times (K+4/3G)/α2 (or (K+4/3G)n) in the simulation (see Section 1.7.1,Eq. (1.102) and Section 1.8.5 for an example).

1.5.3.4 Fluid Moduli for Drained and Undrained Analyses

In FLAC3D, whenever the fluid bulk modulus (or Biot modulus) is selected and CONFIG fluid isspecified, the drained bulk modulus must be specified for the solid matrix. The apparent (undrained)bulk modulus of the solid matrix will then be computed, and will evolve with time.

An undrained analysis may also be performed without specifying CONFIG fluid. In this case, theundrained bulk modulus Ku for the solid matrix should be specified. The undrained bulk modulusis

Ku = K + α2M (1.84)

For an incompressible solid constituent: α = 1, M = Kf /n, and this formula becomes

Ku = K + Kf

n(1.85)

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1.5.4 Porosity

Porosity, n, is a dimensionless number defined as the ratio of void volume to total volume of anelement. It is related to the void ratio, e, by

n = e

1 + e(1.86)

The default value of n, if not specified, is 0.5. n should be given as a positive number between 0and 1, but small values (e.g., less than 0.2) should be used with great caution because the apparentstiffness of the pore fluid is proportional to Kf /n. For low values of n, the stiffness may becomevery large in comparison with the stiffness of the solid material, causing the FLAC3D solution totake a very long time to converge. Consider reducingKf in this case (see Sections 1.5.3.3 and 1.7.1for further guidance).

Porosity is used by FLAC3D to calculate the saturated density of the medium and evaluate Biotmodulus in the case when the fluid bulk modulus is given as an input. FLAC3D does not updateporosity during the calculation cycle, since the process is time-consuming and only the slope of thetransient response is affected. Porosity is a zone property specified using the PROPERTY command.

1.5.5 Saturation

Saturation, s, is defined as the ratio of pore volume occupied by fluid to total pore volume. InFLAC3D’s formulation, pore pressure is set to zero if the saturation at any point is less than 1.The effect of dissolved and trapped air may be allowed by reducing the local fluid modulus whilekeeping the saturation at 1 (i.e., we imagine that there is an equivalent fluid present throughout thepore space). Although no pore pressures are present in a partially saturated region, the trapped fluidstill has weight (i.e., body forces act), and the fluid moves under the action of gravity (at a reducedapparent permeability – see Eq. (1.12)).

The initial saturation may be given by the user, but it is also updated during FLAC3D’s calculationcycle as necessary to preserve the mass balance. Note that in FLAC3D, saturation is not consideredas an independent variable; it cannot be fixed at any gridpoint.

1.5.6 Undrained Thermal Coefficient

The undrained thermal coefficient, β, is defined as the pore-pressure variation divided by 3αM perunit temperature change in an undrained constrained test (no deformation).

For an ideal porous material, this coefficient is related to the volumetric thermal-expansion coeffi-cients for the grains, βg , and fluid, βf , through the formula

β = 3[βg(α − n)+ βf n] (1.87)

The undrained thermal coefficient is a zone property, specified using the PROPERTY command.

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1.5.7 Fluid Tension Limit

In fine soils, the pore water may be able to sustain a significant tension. In FLAC3D, negative porepressures can develop, by default, up to a limit beyond which desaturation will take place. Thenegative pore pressure limit is set using the INITIAL ftens command, where the value is a negativenumber (set to −1015, by default). Note that a negative pore pressure is not the same as “tension”due to capillary, electrical or chemical forces. The latter forces are best represented by an increasedeffective stress within the constitutive model. Negative fluid pressures are unrelated to the fact thata material is composed of grains – the pressures simply arise from the expansion of a volume filledwith fluid.

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1.6 Fluid-Flow Boundary Conditions, Initial Conditions, Sources and Sinks

Boundaries are impermeable by default; all gridpoints are initially “free” (i.e., the pore pressure atsuch gridpoints is free to vary according to the net inflow and outflow from neighboring zones). Thiscondition may be declared explicitly by using the command FREE pp at the appropriate gridpoints.The opposite condition, FIX pp, may also be set at any gridpoint. In general, fluid may enter orleave the grid at an external boundary if the pore pressure is fixed. The effect of these two boundaryconditions is summarized:

1. Pore-pressure free – This is an impermeable boundary and is the default con-dition. There is no exchange between the grid and the outside world. Pressureand saturation changes are computed according to Eq. (1.69) or Eq. (1.70), de-pending on the current value of saturation and whether the fluid has “cavitated”(fallen below the tensile limit).

2. Pore-pressure fixed – This is a permeable boundary across which fluid flowsto and from the outside world. The saturation may only vary if the set pressureis exactly zero. Otherwise, saturation is forced to be 1 (to conform withFLAC3D’s assumption that pore pressures can only exist in a fully saturatedmaterial). Pore pressure cannot be fixed at a value less than the tensile limit;FLAC3D will reset any such pressures to the tensile limit.

As noted above, certain combinations of conditions are impossible (e.g., pore pressure fixed at avalue less than the tensile limit). FLAC3D “corrects” such conditions before executing any calcula-tion step. Pore pressure is fixed at some pressure by using the FIX pp command. Alternatively, thecommand APPLY pp can be used for either external or internal boundaries. It is important to notethat the optional keyword interior must be specified if the condition is to be applied at a non-surfacenode. The use of APPLY has the advantage that the pressure may be controlled directly by a history(supplied by a FISH function).

Fluid-flow boundary conditions can be applied to individual, or ranges of, gridpoints, zone faces orzones via the APPLY command. The APPLY pwell command causes a prescribed inflow or outflowto be applied to a boundary gridpoint. If the interior keyword is added, the condition can be appliedto an interior gridpoint. The APPLY discharge and APPLY leakage commands specify either a fluidflux or a leaky boundary condition to faces of boundary zones. The APPLY vwell command appliesa volume rate of flow to zones within a specified range. All of these fluid-flow boundary conditionsexcept APPLY leakage can refer to a history (via the history keyword). See Section 1.10 for a moredetailed description and format for these commands.

Fixed-pressure gridpoints may act as a source or sinks. There is no explicit command that can beused to measure the inflow or outflow at these points. However, the FISH grid variable gp flowrecords unbalanced nodal flows: a simple function that allows inflows and outflows to be printedor plotted for any range of gridpoints can be written.

Initial distributions of pore pressure, porosity, saturation and fluid-flow properties may be specifiedwith the INITIAL or PROPERTY command, as noted in Section 1.10. If gravity is also given (withthe SET gravity command), it is important that the initial distributions are consistent with thegravitational gradient implied by the value of gravity, the given density of water and the values of

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saturation and porosity within the grid. If the initial distributions are inconsistent, then flow mayoccur in all zones at the start of a run. When setting up a simulation, a few steps should be taken tocheck for this possibility. Section 3.4.7 in the User’s Guide provides a detailed description of theprocedure for setting up initial conditions.

If a model contains interfaces, effective stresses will be initialized along the interfaces (i.e., thepresence of pore pressures will be accounted for within the interface stresses when stresses areinitialized in the grid). This occurs either with or without CONFIG gw specified. For example, theWATER table command (in non-CONFIG gw mode) will include pore pressures along the interface,because pore pressures are defined at gridpoints for interpolation to interface nodes for this mode.Note that flow takes place, without resistance, from one surface to the other surface of an interface,if they are in contact. Preferential flow along an interface (e.g., fracture flow) is not computed.

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1.7 Solving Flow-Only and Coupled-Flow Problems

FLAC3D has the ability to perform both flow-only and coupled fluid-mechanical analyses. Coupledanalysis may be performed with any of the built-in mechanical models in FLAC3D.

Several modeling strategies are available to approach the coupled processes. One of these assumesthat pore pressures once assigned to the grid do not change; this approach does not require anyextra memory to be reserved for flow calculation. The commands associated with this mode arediscussed in Section 1.4.1. Modeling strategies involving flow of fluid require that the CONFIG fluidcommand be issued to configure the grid for fluid analysis, and the MODEL fluid command be issuedfor all zones in which fluid flow can occur. Note that for coupled analysis, the fluid-flow modelis not made null automatically for zones that are made null mechanically. The MODEL fluid fl nullcommand must be given for fluid null zones. The commands associated with CONFIG fluid modeare discussed in Section 1.4.2.

The different modeling strategies for coupled analysis will be illustrated in the following sections,the more elaborate requiring more computer memory and time. As a general rule, the simplestpossible option should be used, consistent with the reproduction of the physical processes that areimportant to the problem at hand. Recommended guidelines for selecting an approach based on timescales are given in Section 1.7.2, and various modeling strategies are described in Sections 1.7.3 to1.7.6.

1.7.1 Time Scales

When planning a simulation involving fluid flow or coupled flow calculations with FLAC3D, it is oftenuseful to estimate the time scales associated with the different processes involved. Knowledge of theproblem time scales and diffusivity help in the assessment of maximum grid extent, minimum zonesize, timestep magnitude and general feasibility. Also, if the time scales of the different processes arevery different, it may be possible to analyze the problem using a simplified (uncoupled) approach.(This approach for fully coupled analyses is discussed in detail in Section 1.7.2.)

Time scales may be appreciated using the definitions of characteristic time given below. Thesedefinitions, derived from dimensional analysis, enter the expression of analytical continuous sourcesolutions. They can be used to derive approximate time scales for the FLAC3D analysis.

Characteristic time of the mechanical process

tmc =√

ρ

Ku + 4/3GLc (1.88)

whereKu is undrained bulk modulus,G is shear modulus, ρ is mass density, andLc is characteristiclength (i.e., the average dimension of the medium).

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Characteristic time of the diffusion process

tfc = L2

c

c(1.89)

where Lc is the characteristic length (i.e., the average length of the flow path through the medium),and c is the diffusivity defined as mobility coefficient k divided by storativity S:

c = k

S(1.90)

There are different forms of storativity that apply in FLAC3D, depending on the controlling process:

1. fluid storage

S = 1

M(1.91)

2. phreatic storage

S = 1

M+ n

ρwgLp(1.92)

3. elastic storage

S = 1

M+ α2

K + 4G/3(1.93)

where M is Biot modulus, α is Biot coefficient (M = Kf /n and α = 1 forincompressible grains), K is drained bulk modulus, G is shear modulus, ρwis fluid density, g is gravity and Lp is characteristic storage length (i.e., theaverage height of the medium available for fluid storage).

For saturated flow-only calculations (rigid material), S is fluid storage and c is the fluid (fromEqs. (1.90) and (1.91)):

c = kM (1.94)

For unsaturated flow calculations, S is phreatic storage and the diffusivity (from Eqs. (1.90) and(1.92)) is

c = k

1/M + n/(ρwgLp)(1.95)

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For a coupled, saturated, deformation-diffusion analysis with FLAC3D, S is elastic storage and c isthe true or generalized coefficient of consolidation, defined from Eqs. (1.90) and (1.93) as

c = k

1M

+ α2

K+4G/3

(1.96)

There are some noteworthy properties based on the preceding definitions:

1. Because explicit timesteps in FLAC3D correspond to the shortest time neededfor the information to propagate from one gridpoint to the next, the magnitudeof the timestep can be estimated using the smallest zone size for Lc in theformula for characteristic time. It is important to note that the explicit fluidflow timestep in FLAC3D is calculated using the fluid diffusivity, Eq. (1.94),even in a coupled simulation. The timestep magnitude may thus be estimatedfrom the formula obtained after substitution of Eq. (1.94) in Eq. (1.89), andusing the smallest zone size Lz for Lc:

t = min

(

L2z

kM

)

(1.97)

2. In a saturated fluid-flow problem, a reduced bulk modulus (i.e.,M orKf ) leadsnot only to an increased timestep, but also to an increased time to reach steadystate, so that the total number of steps, nt , stays the same. This number maybe estimated by taking the ratio of the characteristic times for the model, tc, tothe critical timestep, t , using Eqs. (1.89), (1.94) and (1.97), which gives

nt =(

Lc

Lz

)2

(1.98)

where Lc and Lz are characteristic length for the model and the smallest zone.

3. In a partially saturated fluid-flow problem, adjustments can be made to thefluid modulus (M orKf ) to speed convergence to steady state. Be careful notto reduce M (or Kf ) so much that numerical instability results. A conditionfor stability may be derived from the requirement that the fluid storage (used inthe critical timestep evaluation) must remain smaller than the phreatic storageover one zone height, Lz:

M > aLzρwg/n (1.99)

where a is an adjustment factor chosen equal to 0.3.

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4. Using Eq. (1.89), we see that, to avoid any boundary effects in diffusion prob-lems, the characteristic length,Lc, of the model must be larger than the dimen-sion

√cts , where ts is the maximum simulation time and c is the controlling

diffusivity. In turn, the minimum simulation time is controlled by the relationtmin > L2

z/c.

5. In a coupled flow problem, the true diffusivity is controlled by the stiffnessratio Rk (i.e., the stiffness of the fluid versus the stiffness of the matrix):

Rk = α2M

K + 43G

(1.100)

With this definition for Rk , Eq. (1.96) may be expressed in two forms:

c = kM1

1 + Rk(1.101)

and

c = k

α2

(

K + 4

3G

)

1

1 + 1Rk

(1.102)

If Rk is small (compared to 1), Eq. (1.101) shows that FLAC3D’s standardexplicit timestep (see Eq. (1.97)) can be considered as representative of thesystem diffusivity. An order-of-magnitude estimate for the number of stepsneeded to reach full consolidation under saturated conditions, for instance, canbe calculated using Eq. (1.98).

If Rk is large (i.e., M is large compared to (K + 4G/3)/α2 (or Kf >>>(K+4G/3)n)), FLAC3D’s explicit timestep will be very small, and the problemdiffusivity will be controlled by the matrix (see Eqs. (1.97) and (1.102)). Thevalue forM (orKf ) can be reduced in order to increase the timestep and reachsteady-state computationally faster. Eq. (1.102) indicates that if M (or Kf )is reduced such that Rk = 20, then the diffusivity should be within 5% of thediffusivity for infiniteM (orKf ) (see Section 1.8.5 for an example). The timescale is expected to be respected within the same accuracy. Note that, in anycase, Kf should not be made higher than the physical value of the fluid (2 ×109 Pa for water).

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1.7.2 Selection of a Modeling Approach for a Fully Coupled Analysis

A fully coupled quasi-static hydromechanical analysis with FLAC3D is often time-consuming, andeven sometimes unnecessary. There are numerous situations in which some level of uncouplingcan be performed to simplify the analysis and speed the calculation. The following examplesillustrate the implementation of FLAC3D modeling approaches corresponding to different levels offluid/mechanical coupling. Three main factors can help in the selection of a particular approach:

1. the ratio between simulation time scale and characteristic time of the diffusionprocess;

2. the nature of the imposed perturbation (fluid or mechanical) to the coupledprocess; and

3. the ratio of the fluid to solid stiffness.

The expressions for characteristic time, tfc , in Eq. (1.89), diffusivity, c, in Eq. (1.96), and thestiffness ratio, Rk , in Eq. (1.100) can be used to quantify these factors. These factors are consideredin detail below, and a recommended procedure to select a modeling approach based on these factorsis given in Section 1.7.2.4.

1.7.2.1 Time Scale

We first consider the time scale factor by measuring time from the initiation of a perturbation. Wedefine ts as the required time scale of the analysis, and tc as the characteristic time of the coupleddiffusion process (estimate of time to reach steady state, defined using Eqs. (1.89) and (1.96)).

Short-term behavior

If ts is very short compared to the characteristic time, tc, of the coupled diffusionprocess, the influence of fluid flow on the simulation results will probably benegligible, and an undrained simulation can be performed with FLAC3D (usingeither a “dry” or “wet” simulation – see Section 1.7.5. The wet simulation iscarried out using CONFIG fluid and SET fluid off). No real time will be involvedin the numerical simulation (i.e., ts <<< tc), but the pore pressure will changedue to volumetric straining if the fluid modulus (M or Kf ) is given a realisticvalue. The footing load simulation in Example 1.1 is an example of thisapproach.

Long-term behavior

If ts >>> tc, and drained behavior prevails at t = ts , then the pore-pressurefield can be uncoupled from the mechanical field. The steady-state pore-pressure field can be determined using a fluid-only simulation (SET fluid on,SET mech off) (the diffusivity will not be representative), and the mechanicalfield can be determined next by cycling the model to equilibrium in mechanicalmode with Kf = 0 (SET mech on, SET fluid off). (Strictly speaking, this

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engineering approach is only valid for an elastic material, because a plasticmaterial is path-dependent.)

Another way to describe the time scale is in terms of undrained or drained response. Undrainedstrictly means that there is no exchange of fluid with the outside world (where outside world meansoutside the sample in a lab test, and other elements in a numerical simulation or a field situation).Drained means that there is a full exchange of fluid with the outside world, which implies that thefluid pressure is able to dissipate everywhere. The words are typically associated with short-termand long-term, respectively, because an undrained test usually can be done quickly, while a drainedtest requires a long time for excess fluid pressures to dissipate. In the field, short-term means thatthere is insignificant migration of fluid, and long-term means that the pressure has stabilized (whichneeds a long time).

Note that in simulations conducted outside of the fluid configuration, or within the fluid configurationbut with fluid turned off, the total stress adjustments due to an imposed pore-pressure change (suchas those resulting from the lowering of the water table) will not be done internally by the code.The pore-pressure increments may, however, be monitored (using a FISH function, for instance)and used to decrement the total normal stresses before cycling to mechanical equilibrium. Thesaturated and unsaturated mass densities will also need to be adjusted if the water table has beenmoved within the grid, and the simulation is conducted outside of the fluid configuration. (SeeSection 1.9.2 for additional information on this topic.)

1.7.2.2 Nature of Imposed Perturbation to the Coupled Process

The imposed perturbation to a fully coupled hydromechanical system can be due to changes in eitherthe fluid flow boundary condition or the mechanical boundary condition. For example, transientfluid flow to a well located within a confined aquifer is driven by the change in pore pressures atthe well. The consolidation of a saturated foundation as a result of the construction of a highwayembankment is controlled by the mechanical load applied by the weight of the embankment. Ifthe perturbation is due to change in pore pressures, it is likely that the fluid flow process can beuncoupled from the mechanical process. This is described in more detail below, and illustrated inSection 1.8.6. If the perturbation is mechanically driven, the level of uncoupling depends on thefluid versus solid stiffness ratio, as described below.

1.7.2.3 Stiffness Ratio

The relative stiffness, Rk (see Eq. (1.100)), has an important influence on the modeling approachused to solve a hydromechanical problem:

Relatively stiff matrix (Rk <<< 1)

If the matrix is very stiff (or the fluid highly compressible) and Rk is verysmall, the diffusion equation for the pore pressure can be uncoupled, sincethe diffusivity is controlled by the fluid (Detournay and Cheng 1993). Themodeling technique will depend on the driving mechanism (fluid or mechanicalperturbation):

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1. In mechanically driven simulations, the pore pressure may be assumed toremain constant. In an elastic simulation, the solid behaves as if there wereno fluid; in a plastic analysis, the presence of the pore pressure may affectfailure. This modeling approach is adopted in slope stability analyses (seeSection 1.7.3).

2. In pore pressure-driven elastic simulations (e.g., settlement caused by fluid ex-traction), volumetric strains will not significantly affect the pore-pressure field,and the flow calculation can be performed independently (SET fluid on, SETmech off). (In this case, the diffusivity will be accurate because for Rk <<<1, the generalized consolidation coefficient in Eq. (1.96) is comparable to thefluid diffusivity in Eq. (1.94).) In general, the pore-pressure changes will affectthe strains, and this effect can be studied by subsequently cycling the modelto equilibrium in mechanical mode (SET mech on, SET fluid off). The fluidmodulus (M orKf ) must be set to zero during mechanical cycling, to preventadditional generation of pore pressure.

Relatively soft matrix (Rk >>> 1)

If the matrix is very soft (or the fluid incompressible) and Rk is very large,then the system is coupled, with a diffusivity governed by the matrix. Themodeling approach will also depend on the driving mechanism.

1. In mechanically driven simulations, calculations can be time-consuming. Asdiscussed in note 5 of Section 1.7.1, it may be possible to reduce the value forM (or Kf ), such that Rk = 20, and not significantly affect the response.

2. In most practical cases of pore pressure-driven systems, experience showsthat the coupling between pore pressure and mechanical fields is weak. Ifthe medium is elastic, the numerical simulation can be performed with theflow calculation in flow-only mode (SET fluid on, SET mech off) and then inmechanical-only mode (SET mech on, SET fluid off with fluid modulus set tozero) to bring the model to equilibrium.

It is important to note that, in order to preserve the true diffusivity (and hencethe characteristic time scale) of the system, the fluid modulusM (orKf ) mustbe adjusted to the value

Ma = 11M

+ α2

K+ 43G

(1.103)

or

Kaf = n

nKf

+ 1K+4G/3

(1.104)

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during the flow calculation (see Eq. (1.96)), and to zero during the mechanicalcalculation to prevent further adjustments by volumetric strains (Berchenko1998).

1.7.2.4 Recommended Procedure to Select a Modeling Approach

It is recommended that the selection of a modeling approach for a fully coupled analysis follow theprocedure outlined in Table 1.2. First, determine the characteristic time of the diffusion processfor the specific problem conditions and properties (see Section 1.7.1), and compare this time to theactual time scale of interest. Second, consider whether the perturbation to the system is primarilypore-pressure driven or mechanically driven. Third, determine the ratio of the stiffness of the fluidto the stiffness of the solid matrix. Table 1.2 indicates the appropriate modeling approach based onthe evaluation of these three factors. The table also indicates the required adjustment to the fluidmodulus, Ma (or Ka

f ), for each case. Finally, the table lists several examples from the manual thatillustrate each modeling approach.

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Table 1.2 Recommended procedure to select a modeling approach fora fully coupled analysis

Time Scale Imposed Process Fluid vs Solid Modeling Approach & Adjusted Fluid Examples (6)

Perturbation Stiffness Main Calculation Commands Bulk Modulus (Ma

orKa

f)

ts >>> tc mechanical or any Rk Effective Stress (1) no fluid E.A. 1 (slope3d.f3dat)

(steady-state pore pressure with no fluid flow

analysis) or

Effective Stress (2) Ma

= 0.0

CONFIG fl (orKa

f= 0.0)

SET fluid off

SET mech on

ts <<< tc mechanical or any Rk Pore-Pressure realistic value Example 1.1

(undrained pore pressure Generation (3) forMa

(orKa

f) V.P. 7 (cam.f3dat)

analysis) CONFIG fl

SET fluid off

SET mech on

ts in the pore pressure any Rk Uncoupled Flow- Example 1.16

range of tc Mechanical (4)

CONFIG fl

Step 1. Ma

=1

1M

+ α2K+4G/3

SET fluid on (orKa

f=

n

n

Kf+ 1

K+4G/3)

SET mech off

Step 2. Ma

= 0.0

SET fluid off (orKa

f= 0.0)

SET mech on

ts in the mechanical any Rk Coupled Flow- adjustMa

(orKa

f) Example 1.2

range of tc Mechanical (5) so that Example 1.14

CONFIG fl Rk ≤ 20 Example 1.19

SET fluid on

SET mech on

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Notes to Table 1.2:

1. The effective stress approach with no fluid flow is discussed in Section 1.7.3.In order to establish the initial conditions for this effective stress analysis,use the WATER table command, INITIAL pp command or a FISH function toestablish steady-state pore pressures. Specify the correct wet density to zonesbelow the water table, and dry density to zones above.

2. The effective stress approach with groundwater flow is discussed in Sec-tion 1.7.4. In order to establish the initial conditions for this effective stressanalysis, use the INITIAL command or a FISH function to establish steady-statepore pressures, or specify SET fluid on mech off and step to steady state if thelocation of the phreatic surface is not known. SetMa (orKa

f ) to a small valueto speed convergence for a partially saturated system. Note that Ma (or Ka

f )should be greater than 0.3nLzρwg (or 0.3Lzρwg) to satisfy numerical stability(see Eq. (1.99)).

3. The pore-pressure generation approach is discussed in Section 1.7.5. In orderto establish the initial conditions for the pore-pressure generation analysis,use the INITIAL command or a FISH function to establish steady-state flow,or specify SET fluid on mech off and step to steady state if the location ofthe phreatic surface is not known. Set Ma (or Ka

f ) to a small value to speedconvergence for a partially saturated system. Note that Ma (or Ka

f ) shouldbe greater than 0.3nLzρwg (or 0.3Lzρwg) to satisfy numerical stability (seeEq. (1.99)) .

4. The uncoupled fluid-mechanical approach is described in Section 1.7.2. Thisapproach is recommended for pore pressure-driven systems and should be usedcarefully if Rk >>> 1. Note that the adjusted value for Ma (or Ka

f ) duringthe flow-only step should satisfy Eq. (1.103) so that the coupled diffusivitywill be correct.

5. The fully coupled approach is discussed in Section 1.7.6. Note that forRk >>> 1, if Ma (or Ka

f ) is adjusted to reduce Rk = 20, the time responsewill be close (typically within 5%) to that for infinite M (or Kf ).

6. Example Applications (E.A.) and Verification Problems (V.P.) that demon-strate the various methods are provided in the Examples volume and in theVerifications volume.

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1.7.3 Fixed Pore Pressure (Used in Effective Stress Calculation)

In some calculations, the pore-pressure distribution is important only because it is used in thecomputation of effective stress at all points in the system. For example, in modeling slope stability,we may be given a fixed water table. To represent this system with FLAC3D, it is sufficient to specifya pore-pressure distribution that is unaffected by mechanical deformations that may occur. Becauseno change in pore pressures is involved, we do not need to configure the grid for groundwater flow.

The WATER command can be used to specify an initial hydrostatic pore-pressure distribution belowa given fixed phreatic surface. The water density must be provided (with the WATER densitycommand), and appropriate dry and saturated material densities supplied by the user above andbelow the water table, respectively. Alternatively, the INITIAL pp command or a FISH function canbe used to generate the required static pore-pressure distribution.

The pore-pressure distribution corresponds to an initial state for which there is no strain. It remainsconstant and is unaffected by mechanical deformation. Fluid flow does not take place. The influenceof this pore-pressure distribution is on failure in material for which yield depends on mean effectivestress. For an example application, see Section 1 in the Examples volume.

1.7.4 Flow-Only Calculation to Establish a Pore-Pressure Distribution

Flow-only calculations may be performed to determine the flow and pressure distribution in somesystem independent of any mechanical effects. For example, it may be necessary to evaluatethe groundwater changes that result from the digging of a drainage ditch or the activation of apumping well. In other instances, an initial pore-pressure distribution may be needed for a coupledcalculation. In both cases, FLAC3D may be run in the fluid-flow mode without any mechanicalcalculation being done. Mechanical calculations may or may not be done subsequently.

The first step in the command procedure for a flow-only calculation is to issue a CONFIG fluidcommand so that extra memory can be assigned for the fluid-flow calculation. The mechanicalcalculation should be inhibited with the SET mech off command. Then a choice must be madebetween the explicit and implicit fluid-flow solution algorithm. By default, the explicit algorithmwill be selected, but the implicit mode of calculation may be activated (and deactivated) at anystage of the calculation using the SET fluid implicit on or SET fluid implicit off command. Notehowever, that the implicit calculation mode only applies to problems in which the medium remainsfully saturated (i.e., the saturation remains constant and equal to one); it will give wrong results ifdesaturation occurs in the simulation. When using the implicit mode of calculation, always makesure that desaturation has not taken place (e.g., by plotting saturation contours).

In the explicit mode, the fluid-flow timestep will be calculated automatically, but a smaller timestepcan be selected using the SET fluid dt command. The magnitude of the timestep must be specifiedby the user in the implicit mode. This is done by issuing a SET fluid dt command. For saturatedflow, it is often more efficient to use the implicit solution mode when contrasting permeabilitiesexist.

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The fluid-flow model and properties must be specified for all zones in which fluid flow may occur.Initial and boundary conditions are assigned to complete the fluid-flow problem setup. The fluid-flow domain in a fluid-only or fluid-mechanical simulation is defined by the assembly of zoneswith a non-null fluid flow model. Flux boundary conditions, for instance, can be assigned byspecifying the APPLY command with the range keyword to correspond to the boundaries of thatdomain. (Remember that zones that are nulled mechanically are not automatically nulled for fluidflow.)

The STEP command may be specified to execute a given number of fluid-flow steps. To stop thecalculation automatically when a particular fluid-flow time is reached, a SOLVE age command maybe issued. Alternatively, the setting SET fluid age or SET fluid step may be specified to prescribea fluid-flow time limit or maximum number of steps; then a SOLVE command may be given. Asteady-state flow condition can also be calculated by using the SOLVE command with the ratiokeyword to specify the limiting unbalanced fluid-flow ratio defining the steady flow state.

If the computed pore-pressure distribution is then to be used in a mechanical calculation where porepressure can be assumed to remain constant, the SET fluid off mech on command should be given.The Biot coefficient or fluid bulk modulus should also be set to zero to prevent extra pore pressuresfrom being generated by mechanical deformation.

Section 1.8.3 presents an example flow-only analysis in which unconfined flow through a verticalembankment is calculated. The analysis produces the steady-state phreatic surface. Note that inthis example, the fluid bulk modulus is set to a low value to enable the phreatic surface to developquickly. This can be done when the time taken to reach steady state is unimportant. Note, however,that there is a lower limit for fluid modulus in order to avoid numerical instability (see Eq. (1.99)).

1.7.5 No Flow – Mechanical Generation of Pore Pressure

The undrained (short-term) response may be analyzed in FLAC3D using both dry and wet approaches.In a dry simulation, the generation of pore pressure under volumetric strain is not simulated directly.Instead, its effect on mechanical deformation is taken into account by assigning the undrained bulkvalue,Ku = K+α2M , to the material bulk modulus,K , in the FLAC3D simulation. In this case, twodifferent techniques can be applied to detect failure in a Mohr-Coulomb material. In the first one,the constant pore pressures are initialized using the WATER or INITIAL pp command, and undrainedcohesion and friction are given as input. In the second, the material is assigned a zero friction and acohesion equal to the undrained shear strength, Cu. The first technique applies to problems wherechanges in pore pressure are small compared to the initial values. The second technique strictlyapplies to plane-strain problems with Skempton coefficient equal to one (M >>> K + 4G/3).Note that a dry simulation can be carried out whether or not the CONFIG fluid command has beenissued. However, if the command has been used, the fluid bulk modulus (M or Kf ) must be set tozero to prevent additional generation of pore pressure.

In a wet simulation, the short-term response of a coupled system is analyzed in the fluid configurationof FLAC3D. In this case, drained values should be used for the material bulk modulus, friction andcohesion. If the SET fluid off command is given and the Biot modulus (or fluid modulus) is givena realistic value, then pore pressure will be generated as a result of mechanical deformations. For

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example, the “instantaneous” pore pressures produced by a footing load can be computed in thisway. If the fluid bulk modulus (M orKf ) is much greater than the solid bulk modulus, convergencewill be slow; it may be possible to reduce the fluid bulk modulus without significantly affecting thebehavior. (See Note 5 in Section 1.7.1.) The data file in Example 1.1 (“instpp.f3dat”) illustratespore pressure build-up produced by a footing load on an elastic/plastic material contained in a box.The corresponding project file, “footing.f3prj,” is located in folder “datafiles\fluid\footing.” Theleft boundary of the box is a line of symmetry, and the pore pressure is fixed at zero along the topsurface to prevent pore-pressure generation there. By default, the porosity is 0.5; permeability isnot needed, since flow is not calculated.

As a large amount of plastic flow occurs during loading, the normal stress is applied graduallyby using the FISH function ramp to supply a linearly varying multiplier to the APPLY command.Figure 1.2 shows pore-pressure contours and vectors representing the applied forces. It is importantto realize that the plastic flow will occur in reality over a very short period of time (on the order ofseconds); the word “flow” here is misleading since, compared to fluid flow, it occurs instantaneously.Hence, the undrained analysis (with SET fluid=off) is realistic.

Example 1.1 Adding load

newtitle ’Instantaneous pore pressures generated under an applied load’config fluidgen zone brick size 20 1 10; --- mechanical model ---model mech mohrprop bulk 5e8 shear 3e8 fric 25 coh 1e5 tens 1e10fix x range x -.1 .1fix z range z -.1 .1fix x y z range x 19.9 20.1fix y; --- apply load slowly ---def ramp

ramp = min(1.0,float(step)/200.0)endapply nstress = -0.5e6 hist @ramp range x -.1 3.1 z 9.9 10.1; --- fluid flow model ---model fluid fl_isoini fmod 2e9; --- pore pressure fixed at zero at the surface ---fix pp 0 range z 9.9 10.1; --- settings ---set fluid off; --- histories ---hist add gp pp 2,.5,9; --- test ---step 750

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save loadhist dump 1 begin 750return

Figure 1.2 Instantaneous pore pressures generated under an applied load

1.7.6 Coupled Flow and Mechanical Calculations

By default, FLAC3D will do a coupled fluid-flow and mechanical calculation if the grid is configuredfor fluid, and if the Biot modulus, or fluid bulk modulus, and permeability are set to realistic values.The full fluid-mechanical coupling in FLAC3D occurs in two directions: pore-pressure changescause volumetric strains to occur that influence the stresses; in turn, the pore pressure is affectedby the straining that takes place.

The relative time scales associated with consolidation and mechanical loading should be appreciated.Mechanical effects occur almost instantaneously (on the order of seconds or fractions of seconds).However, fluid flow is a long process: the dissipation associated with consolidation takes place overhours, days or weeks.

Relative time scales may be estimated by considering the ratios of characteristic times for the coupledand undrained processes. The characteristic time associated with the undrained mechanical processis found by using the saturated mass density for ρ and the undrained bulk modulus, Ku, as definedin Eq. (1.84) (or Eq. (1.85) for incompressible grains) forK in Eq. (1.88). The ratio of the diffusioncharacteristic time (Eqs. (1.89) and (1.96)) and the undrained mechanical characteristic time is thus

FLAC3D Version 4.0


tfc

tmc=√

K + α2M + 4/3G

ρ

Lc

k

(

1

M+ α2

K + 4/3G

)

(1.105)

In most cases, M is approximately 1010 Pa, but the mobility coefficient k may differ by severalorders of magnitude; typical values are:

10−19m2/Pa-sec for granite;

10−17m2/Pa-sec for limestone;

10−15m2/Pa-sec for sandstone;

10−13m2/Pa-sec for clay; and

10−7m2/Pa-sec for sand.

For rock and soil, ρ is of the order of 103 kg/m3, whileK + 4/3G is approximately 1010 Pa. Usingthose orders of magnitude in Eq. (1.105), we see that the ratio of fluid-to-mechanical time scalesmay vary between Lc for sand, 106Lc for clay, 108Lc for sandstone, 1010Lc for limestone and1012Lc for granite. If we exclude materials with mobility coefficients larger than that of clay, wesee that this ratio remains very large, even for small values of Lc.

In practice, mechanical effects can then be assumed to occur instantaneously when compared todiffusion effects. This is also the approach adopted in FLAC3D, where no time is associated withany of the mechanical steps taken together with the fluid-flow steps. The use of the dynamic optionin FLAC3D may be considered, to study the fluid-mechanical interaction in materials such as sand,where mechanical and fluid time scales are comparable.

In most modeling situations, the initial mechanical conditions correspond to a state of equilibriumthat must first be achieved before the coupled analysis is started. Typically, at small fluid-flowtime (compared to tc – see Section 1.7.1), a certain number of mechanical steps must be taken foreach fluid step to reach quasi-static equilibrium. At larger fluid-flow time, if the system approachessteady-state flow, several fluid-flow timesteps may be taken without significantly disturbing themechanical state of the medium. A corresponding numerical simulation may be controlled manuallyby alternating between flow-only (SET fluid on mech off) and mechanical-only (SET fluid off mechon) modes. The STEP command can be used to perform calculations for both the flow-only andmechanical-only processes.

As an alternative approach, the tedious task of switching between flow-only and mechanical-onlymodes may be avoided by using the SOLVE command in combination with appropriate settings;SET mech force (or SET mech ratio) will set a limit to the out-of-balance force (or force ratio), underwhich quasi-static mechanical equilibrium will be assumed. SET mech substep n auto declares themechanical module as the slave that must perform n sub-cycles (or fewer, if equilibrium is detected– see previous setting) for each step taken by the master. SET fluid substep m implicitly declaresthe flow module as the master. (The keyword auto is omitted for the master process.) If for eachfluid-flow timestep the mechanical module needs only one sub-step to reach equilibrium, then the

FLAC3D Version 4.0


number of fluid-flow sub-steps is doubled, but never exceeds the number m (by default, m = 1).The system reverts to the original setting of sub-steps whenever the succession of one mechanicaltimestep for each fluid-flow group of sub-steps is broken. If the option age is specified in the SOLVEcommand, the computation will proceed until the fluid-flow time given by the age parameter isreached.

In a third approach, the STEP command may be used while both mechanical and fluid-flow modulesare on. In this option, one mechanical step will be taken for each fluid-flow step. Here, fluid-flowsteps are assumed to be so small that one mechanical step is enough to re-equilibrate the systemmechanically after each fluid-flow step is taken. Section 1.10.1 should be consulted for a completelist of available command options for a coupled analysis.

To illustrate a fully coupled analysis, we continue the footing simulation done in Section 1.7.5, usingthe SOLVE command with appropriate settings to calculate the consolidation beneath the footing.The data file is “coupled.f3dat,” shown in Example 1.2:

Example 1.2 A coupled fluid flow-mechanical analysis

restore load; --- turn on fluid flow model ---prop perm 1e-12set fluid onini xvel 0.0 yvel 0.0 zvel 0.0ini xd 0.0 yd 0.0 zd 0.0; --- set mechanical limits ---set mech force 1.5e3set mech subs 100 auto ;slaveset fluid subs 10; --- histories ---hist add fltimehist add gp zd 0,1,10hist add gp zd 1,1,10hist add gp zd 2,1,10hist add gp pp 2,1,9hist add gp pp 5,1,5hist add gp pp 10,1,7; --- solve to 5000 sec ---solve age 0.5e4save age_5k

; --- solve to 150,000 sec ---solve age 1.5e5save age_150kreturn

FLAC3D Version 4.0


The screen printout should be watched during the calculation process – eight variables are updatedon the screen after the SOLVE command is issued: (1) the current step number; (2) the total numberof sub-cycles taken by the master process (fluid flow) for the most recent step; (3) the total numberof sub-cycles taken by the slave process (mechanical) for the most recent step; (4) the process thatis currently active; (5) the current sub-cycle number; (6) the current maximum unbalanced force orforce ratio; (7) the total time for the master (fluid flow) process; and (8) the current timestep.

The force tolerance is 1.5 ×103 for this problem; enough mechanical steps are taken to keep theunbalanced force below this force tolerance. However, the limit to mechanical steps, defined bySET mech substep, is set to 100 in this example. If the actual number of mechanical steps takenis always equal to the set value of SET mech substep, then something must be wrong. Either theforce limit or SET mech substep has been set too low, or the system is unstable and cannot reachequilibrium. The quality of the solution depends on the force tolerance: a small tolerance will givea smooth, accurate response, but the run will be slow; a large tolerance will give a quick answer,but it will be noisy.

The characteristic diffusion time for this coupled analysis is evaluated from Eq. (1.89), usingEq. (1.96) for the diffusivity and a value ofLc = 10 m corresponding to the model height. Using theproperty values in Example 1.1, tfc is estimated at 140,000 seconds. Full consolidation is expectedto be reached within this time scale; the numerical simulation is carried out for a total of 150,000seconds.

The pore-pressure distribution at a time of 5000 seconds is shown in Figure 1.3 (note that the bulbof maximum pore pressure has migrated downward and smoothed out). Figure 1.4 shows thetime histories to 150,000 seconds of displacements under the footing load. In this simulation, porepressures remain fixed at zero on the ground surface; hence, the excess fluid escapes upward. Theleveling off of the histories indicate that full consolidation has been reached.

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Figure 1.3 Pore-pressure distribution at 5000 seconds

Figure 1.4 Consolidation response – time histories of footing displacements

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If a sudden change of loading or mechanical boundary condition is applied in a coupled problem,it is important to allow the undrained (short-term) response to develop before allowing flow to takeplace. In other words, FLAC3D should be run to equilibrium under SET fluid off conditions followingthe imposed mechanical change. The SOLVE logic can then be used (with SET fluid on) to computethe subsequent coupled flow/mechanical response. If changes in fluid boundary conditions occurphysically at the same time as mechanical changes, then the same sequence should be followed(i.e., mechanical changes . . . equilibrium . . . fluid changes . . . coupled solution).

Another example of fully coupled behavior is the time-dependent swelling that takes place followingthe excavation of a trench in saturated soil. In this case, negative pore pressures build up immediatelyafter the trench is excavated; the subsequent swelling is caused by the gradual influx of water into theregion of negative pressures. We model the system in two stages: in the first, we allow mechanicalequilibrium to occur, without flow; then we allow flow, using the SOLVE command to maintain quasi-static equilibrium during the consolidation process. The fluid tension is initialized to a large negativenumber to prevent desaturation. The data file used is “swelling.f3dat,” shown in Example 1.3. Thecorresponding project file is “swelling.f3prj,” located in folder “datafiles\fluid\swelling.”

The trench is excavated in the left-hand part of a flat soil deposit that is initially fully saturatedand in equilibrium under gravity. The material is elastic in this case, but it could equally well havebeen a cohesive material, such as clay. In this run, we assume impermeable conditions for the freesurfaces. Figure 1.5 shows the displacement vectors that accumulate during the time that flow isoccurring; the trench is seen at the left-hand side of the model. Figure 1.6 shows the time history ofpore pressure near the crest of the trench; note that there is an initial negative excursion in pressurearising from the instantaneous expansion of the soil toward the trench. Figure 1.7 shows histories ofhorizontal and vertical displacement at the crest. The characteristic time for this problem, evaluatedusing the model length of 40 m for Lc, is approximately 5 ×108 seconds (based on Eqs. (1.89) and(1.96)); the numerical simulation is carried out to that time.

Example 1.3 Maintaining equilibrium under time-dependent swelling conditions

newtitle "Maintaining equilibrium under time-dependent swelling conditions"

config fluidgen zone brick size 40 1 8; --- mechanical model ---model mech elasprop bulk 2e8 shear 1e8ini dens 1500model mech null range x 0,2 z 2,8; --- fluid flow model ---model fluid fl_isoprop perm 1e-14 poros 0.5ini fmod 2e9ini ftens -5e5ini fdens 1000

FLAC3D Version 4.0


model fluid fl_null range x 0,2 z 2,8; initial and boundary conditionsfix x range x -.1 .1fix x range x 39.9 40.1fix x y z range z -.1 .1fix yset grav 0,0,-10ini sxx -1.6e5 grad 0,0,20000ini syy -1.6e5 grad 0,0,20000ini szz -1.6e5 grad 0,0,20000ini pp 8.0e4 grad 0,0,-10000; --- settings ---set fluid off;hist add unbalsolvesave swell1;ini xd 0.0 yd 0.0 zd 0.0his add fltime;his add gp pp 3,0,7his add zone pp 4.5,0.5,6.5his add gp xd 2,0,8his add gp zd 2,0,8fix pp range x 39.9 40.1set fluid onset mech force 50set fluid substep 100set mech substep 100 auto ;slavesolve age 5e8save swell2return

FLAC3D Version 4.0


Figure 1.5 Swelling displacements near a trench with impermeable surfaces

Figure 1.6 History of pore pressure behind the trench face

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Figure 1.7 Displacement histories at the trench crest – vertical (top) andhorizontal (bottom)

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1.8 Verification Examples

Several verification problems that illustrate the fluid-flow modeling capabilities in FLAC3D arepresented. The data files for these examples are contained in the “\Datafiles\Fluid” directory.

1.8.1 Unsteady Groundwater Flow in a Confined Layer

A long embankment of width L = 100 m rests on a shallow saturated layer of soil. The width (L) ofthe embankment is large in comparison with the layer thickness, and its permeability is negligiblewhen compared to the permeability, k = 10−12 m2/(Pa sec), of the soil. The Biot modulus for thesoil is measured to beM = 10 GPa. Initial steady-state conditions are reached in the homogeneouslayer. The purpose is to study the pore-pressure change in the layer as the water level is raisedinstantaneously upstream by an amount H0 = 2 m. This corresponds to a pore-pressure rise ofp1 = H0ρwg (with the water density ρw = 1000 kg/m3 and acceleration of gravity g = 10 m/s2) atthe upstream side of the embankment. Figure 1.8 shows the geometry of the problem:

embankment

L

z

Figure 1.8 Confined flow in a soil layer

The flow in the layer may be assumed to be one-dimensional. The model has width L. The excesspore pressure, p, initially zero, is raised suddenly to the value p1 at one end of the model. Thecorresponding analytical solution has the form

FLAC3D Version 4.0


p(z, t) = 1 − z− 2

π

∞∑

n=1

e−n2π2 t

(

sin nπz

n

)

(1.106)

where the z-axis is running along the embankment width and has its origin at the upstream side,p = p

p1, z = z

L, t = ct/L2 and c = Mk.

In the FLAC3D model, the layer is defined as a column of 25 zones. The excess pore pressure isfixed at the value of 2 ×104 Pa at the face located at z = 0, and at zero at the face located at z =100 m. The model grid is shown in Figure 1.9.

The analytical solution is programmed as a FISH function for direct comparison to the numericalresults at selected fluid-flow times corresponding to t = 0.05, 0.1, 0.2 and 1.0. The analytical andnumerical pore-pressure results for these times are stored in tables.

Figure 1.9 FLAC3D grid for fluid flow in a confined soil layer

Example 1.4 contains the FLAC3D data file for this problem, using the explicit formulation to obtainthe solution. Example 1.5 contains the data file using the implicit formulation. The correspondingproject file, “ugwFlowConfLayer.f3prj,” is located in folder “datafiles\fluid\ugwFlowConfLayer.”The comparison of analytical and numerical excess pore pressures at four fluid-flow times for theexplicit solution is shown in Figure 1.10, and for the implicit solution in Figure 1.11. Normalizedexcess pore pressure (p/p1) is plotted versus normalized distance (z/L) in the two figures, whereTables 2, 4 and 6 contain the analytical solution for excess pore pressures, and Tables 1, 3 and 5

FLAC3D Version 4.0


contain the FLAC3D solutions. The four flow times are 5 ×104, 105, 2 ×105 and 106 seconds forboth explicit and implicit solutions. Steady-state conditions are reached by the last time considered.For both solution formulations, the difference between analytical and numerical pore pressures atsteady state is less than 0.1%.

Example 1.4 Unsteady groundwater flow in a confined layer – explicit solution

newset fish autocreate off

title ’Unsteady groundwater flow in a confined layer: explicit method’config fluid; --- fish constants ---def constants

global c_cond = 1e-12 ; permeabilityglobal c_biom = 1e10 ; biot modulusglobal length = 100. ; layer widthglobal dp1 = 2e4 ; pore pressure rise, face 1global tabn = -1global tabe = 0global overl = 1. / lengthlocal d = c_cond * (c_biom)global dol2 = d * overl * overlglobal pi2 = pi * pi

end@constantsgen zone brick size 1 1 25 p1 10 0 0 p2 0 10 0 p3 0 0 @length; --- fluid flow model ---model fluid fl_isoset fluid biot onprop perm @c_cond ; default value: poros 0.5 biot_c 1ini biot_mod @c_biom ; could also use: ini fmodulus 0.5e10fix pp @dp1 range z -.1 .1fix pp 0 range z 99.9 100.1; --- fish function ---def num_sol

tabn = tabn + 2local t_hat = fltime * dol2local pnt = gp_headloop while pnt # null

local rad = sqrt(gp_xpos(pnt)ˆ2 + gp_ypos(pnt)ˆ2) * overlif rad < 1.e-4 then

local x = gp_zpos(pnt) * overltable(tabn,x) = gp_pp(pnt) / dp1

end_ifpnt = gp_next(pnt)

FLAC3D Version 4.0


end_looptable_name(tabn) = ’FLAC3D at ’+string(t_hat)

enddef ana_sol

local top = 2. / pilocal n_max = 100 ; max number of terms -exact solutiontabe = tabe + 2local t_hat = fltime * dol2local tp2 = t_hat * pi2local pnt = gp_headloop while pnt # null


local x = gp_zpos(pnt) * overllocal n = 0local nit = 0local tsum = 0.0local tsumo = 0.0local converge = 0loop while n < n_max

n = n + 1local fn = float(n)local term = sin(pi*x*fn) * exp(-tp2*fn*fn) / fntsum = tsumo + termif tsum = tsumo then

nit = ntable(tabe,x) = 1. - x - top * tsumconverge = 1n = n_max

elsetsumo = tsum

end_ifend_loopif converge = 0 then

local str = buildstr("no convergence x = %1 - t = %2",x,fltime)local oo = out(str)exit

end_ifend_ifpnt = gp_next(pnt)

end_looptable_name(tabe) = ’Analytical at ’+string(t_hat)

end; --- settings ---set mech offset fluid on

FLAC3D Version 4.0


; --- test ---solve age 5e4list gp pp range x -.001 .001 y -.001 .001@num_sol@ana_sol

solve age 10e4@num_sol@ana_sol



save confe-imp

Figure 1.10 Comparison of excess pore pressures for the explicit-solution al-gorithm (analytical values = lines; numerical values = crosses)

FLAC3D Version 4.0


Example 1.5 Unsteady groundwater flow in a confined layer – implicit solution


title ’Unsteady groundwater flow in a confined layer: implicit method’config fluid; --- fish constants ---def constants

global c_cond = 1e-12 ; permeabilityglobal c_biom = 1e10 ; biot modulusglobal length = 100. ; layer widthglobal dp1 = 2e4 ; pore pressure rise, face 1global tabn = -1global tabe = 0global overl = 1. / lengthlocal d = c_cond * (c_biom)global dol2 = d * overl * overlglobal pi2 = pi * pi

end@constantsgen zone brick size 1 1 25 p1 10 0 0 p2 0 10 0 p3 0 0 @length; --- fluid flow model ---model fluid fl_isoset fluid biot onprop perm @c_cond ; default value: poros 0.5 biot_c 1ini biot_mod @c_biom ; could also use: ini fmodulus 0.5e10fix pp @dp1 range z -.1 .1fix pp 0 range z 99.9 100.1; --- fish function ---def num_sol

tabn = tabn + 2local t_hat = fltime * dol2local pnt = gp_headloop while pnt # null


local x = gp_zpos(pnt) * overltable(tabn,x) = gp_pp(pnt) / dp1


end_looptable_name(tabn) = ’FLAC3D at ’+string(t_hat)

enddef ana_sol

FLAC3D Version 4.0


local top = 2. / pilocal n_max = 100 ; max number of terms -exact solutiontabe = tabe + 2local t_hat = fltime * dol2local tp2 = t_hat * pi2local pnt = gp_headloop while pnt # null


local x = gp_zpos(pnt) * overllocal n = 0local nit = 0local tsum = 0.0local tsumo = 0.0local converge = 0loop while n < n_max

n = n + 1local fn = float(n)local term = sin(pi*x*fn) * exp(-tp2*fn*fn) / fntsum = tsumo + termif tsum = tsumo then

nit = ntable(tabe,x) = 1. - x - top * tsumconverge = 1n = n_max

elsetsumo = tsum

end_ifend_loopif converge = 0 then

local str = buildstr("no convergence x = %1 - t = %2",x,fltime)local oo = out(str)exit


end_looptable_name(tabe) = ’Analytical at ’+string(t_hat)

end; --- settings ---set mech offset fluid onset fluid implicit onset fluid dt 1e3; --- test ---solve age 5e4

FLAC3D Version 4.0


list gp pp range x -.001 .001 y -.001 .001@num_sol@ana_sol




save confe-exp

Figure 1.11 Comparison of excess pore pressures for the implicit-solutionalgorithm (analytical values = lines; numerical values = crosses)

FLAC3D Version 4.0


1.8.2 One-Dimensional Filling of a Porous Region

In this problem, flow is driven through an initially dry porous layer of large lateral extent under aconstant pressure, po , applied at the base. The transient location of the filling front is compared toan exact sharp-front solution for the cases with and without gravity.

Voller et al. (1996) give an analytic solution for this problem under the assumptions of a sharp-front,rigid-porous matrix and incompressible Newtonian fluid. In their solution, the flow is governed byDarcy’s law, and there is a constant atmospheric pressure in the air ahead of the free surface.

Let the x-axis of reference be oriented in the direction of flow, with the origin at the base of the layer.The solution for the front location, xf , may be expressed in terms of two dimensionless variables:t = t/T ∗ and x = xf /L

∗; and a dimensionless parameter, γ , defined using the expressionsT ∗ = nμ/po, L∗ = √

κ and γ = √κ ρg/po. In these equations, n is porosity, κ is intrinsic

permeability (product of mobility coefficient, k, and dynamic viscosity, μ), ρ is fluid density and gis gravity.

When gravity is ignored, the solution has the form

t = 1

2[x]2 (1.107)

For filling under gravity, the front location is given by

t = − xγ

− ln (1 − γ x)

γ 2(1.108)

Eq. (1.108) may be shown to converge to the no gravity solution when γ goes to zero.

The numerical solution to the filling problem is presented in dimensionless form. To derive thissolution, scaled properties are used in the simulations: po = 1, k = 0.25, n = 0.5; and for gravity flow,ρw = 1 and g = 1. Further, using μ = 4 in the preceding definitions, the characteristic parametersfor the simulation are T ∗ = 2, L∗ = 1, γ = 1.

To simulate incompressible flow, the bulk modulus of the fluid,Kf , is assigned a value that is largecompared to the pore-pressure variations in the simulation (Kf = 100). For both cases, the gridcorresponds to a column of 25 zones, 0.625 units high and 0.025 units wide. The initial value ofpore pressure and saturation is zero. The pore pressure is fixed at po, and the saturation is given avalue of one at the base of the model. The simulation is conducted for a total of 0.25 units of time( t = 0.125). A FISH function, flacfront, captures the times at which nodes reach a saturation of1% and 99%. The analytic sharp-front solution is evaluated by another FISH function, solution.The data file for the case without gravity is listed in Example 1.6; the data file for the case withgravity is listed in Example 1.7. The corresponding project file, “1DFillingPorousReg.f3prj,” islocated in folder “datafiles\fluid\1DFillingPorousReg.”

FLAC3D Version 4.0


Example 1.6 1D filling of a porous region – no gravity


title "One-dimensional filling - no gravity"def setup

global c_perm = 0.25global c_poro = 0.5global c_p0 = 1.global c_visc = 4global ils = 1global ius = 1global dz = .025global lzf = 0.0global uzf = 0.0

end@setupconfig fluidgen zone brick size 1 1 25 p1 .025 0 0 p2 0 .025 0 p3 0 0 0.625; --- fluid flow model ---model fluid fl_isoprop perm @c_perm poros @c_poro biot_c 0. ; default: poros 0.5 biot_c 1set fluid biot onini biot_mod 100ini sat 0.0fix pp @c_p0 range z -.0001 .0001; --- settings ---set mech offset fluid on; --- fish functions ---def solution

global Tstar = c_poro * c_visc / c_p0global Lstar = sqrt(c_perm * c_visc)local kloop k (1,25)

local Xhat = float(k-1) * 0.025 / Lstarlocal That = Xhat * Xhat * 0.5xtable(1,k) = Thatytable(1,k) = Xhat

endloopend@solutiondef flacfront

while_stepping

FLAC3D Version 4.0


local pntl = gp_near(1,1,lzf)local pntu = gp_near(1,1,uzf)if gp_sat(pntl) > .01 then

xtable(2,ils) = fltime / Tstarytable(2,ils) = lzf / Lstarils = ils + 1lzf = lzf + dz

endifif gp_sat(pntu) > .99 then

xtable(3,ius) = fltime / Tstarytable(3,ius) = uzf / Lstarius = ius + 1uzf = uzf + dz

endifendtable 1 name ’Analytical solution’table 2 name ’ 1 saturation front’table 3 name ’99% saturation front’; --- test ---set @lzf = .025 @uzf = .025solve age 0.25save asat1return

Example 1.7 1D filling of a porous region – with gravity


title "One-dimensional filling - with gravity"def setup

global c_perm = 0.25global c_poro = 0.5global c_p0 = 1.global c_visc = 4global c_den = 1.global c_grav = -1.global ils = 1global ius = 1global dz = .025global lzf = 0.0global uzf = 0.0

end@setupconfig fluid

FLAC3D Version 4.0


gen zone brick size 1 1 25 p1 .025 0 0 p2 0 .025 0 p3 0 0 0.625; --- fluid flow model ---model fluid fl_isoprop perm @c_perm poros @c_poro biot_c 0. ; default: poros 0.5 biot_c 1set fluid biot onini biot_mod 100ini sat 0.0fix pp @c_p0 range z -.0001 .0001; --- settings ---ini fdensity @c_denset gravity 0 0 @c_gravset mech offset fluid on; --- fish functions ---def solution

global Tstar = c_poro * c_visc / c_p0global Lstar = sqrt(c_perm * c_visc)local gamma = -Lstar * c_den * c_grav / c_p0local kloop k (1,50)

local Yhat = ((k-1)*0.7/49.) / Lstarlocal That = - (Yhat + ln(1.0-gamma * Yhat) / gamma) / gammaxtable(1,k) = Thatytable(1,k) = Yhat

endloopend@solutiondef flacfront

while_steppinglocal pntl = gp_near(1,1,lzf)local pntu = gp_near(1,1,uzf)if gp_sat(pntl) > .01 then

xtable(2,ils) = fltime / Tstarytable(2,ils) = lzf / Lstarils = ils + 1lzf = lzf + dz

endifif gp_sat(pntu) > .99 then

xtable(3,ius) = fltime / Tstarytable(3,ius) = uzf / Lstarius = ius + 1uzf = uzf + dz

endifendtable 1 name ’Analytical solution’table 2 name ’ 1 saturation front’

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table 3 name ’99% saturation front’; --- test ---set @lzf = .025 @uzf = .025solve age 0.25save asat2return

The results with and without gravity are presented in Figure 1.12 and Figure 1.13, respectively.As seen in these figures, the sharp-front solution is bounded above and below by the 99% and 1%saturation fronts. In fact, the vertical distance between these fronts corresponds directly to the gridsize in the direction of propagation of the filling front. (The saturation at a node can only startto increase when the pore pressure at the node below it becomes positive, and thus full saturationis reached there.) This distance can be reduced by increasing the number of zones in the columnheight. The evolution of nodal pore pressure with time follows a stepwise pattern, more pronouncedas the fluid is less compressible. This behavior occurs because a node must be fully saturated beforeits pore pressure can increase. One way to reduce this effect without changing the grid size is tointroduce flow in the unsaturated region (capillary pressure) in the fluid flow formulation.

Figure 1.12 Location of filling front ( x vs t ) – no gravity(analytical solution = solid line; numerical values = dashed line)

FLAC3D Version 4.0


Figure 1.13 Location of filling front ( x vs t ) – with gravity(analytical solution = solid line; numerical values = dashed line)

1.8.3 Steady-State Fluid Flow with a Free Surface

This example is the classical problem of steady-state seepage flow through a homogeneous em-bankment with vertical slopes exposed to different water levels and resting on an impermeablebase. The total discharge,Q, and the length of seepage face, s, are compared to the exact solutions.Figure 1.14 shows the geometry and boundary conditions of the problem.

FLAC3D Version 4.0


��

��

�

��

�

Figure 1.14 Problem geometry and boundary conditions

The fluid is homogeneous, flow is governed by Darcy’s law, and it is assumed that the pores ofthe soil beneath the phreatic surface are completely filled with water, and the pores above it arecompletely filled with air. The width of the dam is L, the head and tail water elevations above theimpervious base are h1 and h2, respectively.

The exact solution for the total discharge through a dam section of unit thickness was shown byCharny (Harr 1991) to be given by Dupuit’s formula:

Q = k ρw g · h21 − h2

2

2L(1.109)

where k is mobility coefficient, ρw is water density and g is gravity. The length, s, of the seepageface (elevation of the free surface on the downstream face of the dam above h2) was obtainedby Polubarinova-Kochina (1962), and is given in Figure 1.15 as a function of the characteristicdimensions of the problem.

FLAC3D Version 4.0


��

��

��

��

��

��

� ��

0.5

11.5

2.5

33.5

2

Figure 1.15 Seepage face solution after Polubarinova-Kochina (1962)

The FLAC3D simulation is conducted for the following particular set of parameters:

L = 9 mh1 = 6 mh2 = 1.2 m

The following material properties are used:

permeability (k) 10−10 (m/sec)/(Pa/m)porosity (n) 0.3

water density (ρw) 1000 kg/m3

water bulk modulus (Kf ) 1000 Pa

soil dry density (ρ) 2000 kg/m3

gravity (g) 10 m/sec2

Two cases corresponding to two different initial conditions have been studied:

CASE 1: The water level in the embankment is initially at h = h2 = 1.2 m, and theupstream level is raised to h = h1.

CASE 2: The water level is initially at h = h1 = 6 m, and the downstream level is loweredto h = h2.

FLAC3D Version 4.0


The data file for Case 1 is listed in Example 1.8, and for Case 2 in Example 1.9. The correspondingproject file, “steadystateFF-FreeSurface.f3prj,” is located in folder “datafiles\fluid\steadystateFF-FreeSurface.” The grid and boundary conditions are the same for both cases. The grid contains 30zones in width, 20 zones in height and 1 zone in thickness. The boundary conditions correspond to astatic pore-pressure distribution up to level h1 on the upstream face, and up to h2 on the downstreamface, zero pore pressure from level h2 to h1 on the downstream face, and to no flow conditions acrossthe remaining boundaries. The differences between the two cases are the initial pore pressure andsaturation distributions. In Case 1, saturation and pore pressure are zero above h2. Below thatlevel, the saturation is 1 and the pressure is hydrostatic. In Case 2, saturation is 1 for all gridpoints,and the pore pressure inside the mesh follows a gravitational gradient. The numerical simulationis carried out until steady-state conditions are detected.

To speed the calculation to steady state, the water bulk modulus is given a small value (Kf =103 Pa) compatible with free surface stability. (The criterion used is Kf ≥ 0.3ρwgLz, where Lzis the maximum vertical zone dimension in the vicinity of the phreatic surface, as discussed inSection 1.7.1.)

The final flow pattern is similar for both initial conditions (see Figures 1.16 and 1.17). The numericalvalue of seepage length is defined as the distance on the downstream face of the dam, between thetail water elevation and the point where the magnitude of the flow vector vanishes. The analyticalvalue of seepage length is determined from Figure 1.15. For this particular problem, h2/h1 = 0.2,L/h1 = 1.5, and the value of s/h1 is thus 0.1. As seen in the figures, the numerical value of seepagelength compares well with the analytical solution sketched there as a bold line. A FISH function,qflac, is used to determine the discharge, Q, per unit thickness of the dam: the steady-statenumerical value is 1.914 × 10−6m2/s for Case 1; and 1.912 × 10−6m2/s for Case 2. The values areclose to the analytic value of 1.920 × 10−6m2/s, as determined from Eq. (1.109) for this particularproblem.

Example 1.8 Steady-state flow through a vertical embankment – Case 1

newtitle "Steady state flow through a vertical embankment - case 1"def setup

global c_perm = 1e-10global c_poro = 0.3global c_kw = 1e3global c_L = 9.global c_h1 = 6.global c_h2 = 1.2

end@setupconfig fluidgen zone brick size 30 1 20 p1 @c_L 0 0 p2 0 0.15 0 p3 0 0 @c_h1; --- fluid flow model ---model fluid fl_isoprop perm @c_perm poros @c_poro ; default value: poros 0.5 biot_c 1

FLAC3D Version 4.0


set fluid biot offini fmodulus @c_kwini sat 0.0ini sat 1 range z -.1 1.1999ini sat 1 range x -.0001 .0001ini pp 1.2e4 grad 0 0 -1e4 range z -.1 1.1999fix pp range x 8.99 9.01ini pp 6e4 grad 0 0 -1e4 range x -.0001 .0001fix pp range x -.0001 .0001; --- settings ---ini fdensity 1e3ini ftens 0.0set gravity 0 0 -10set mech offset fluid on; --- test ---solve ratio 1.e-3def qflac

local qval = 0.0local pnt = gp_headloop while pnt # null

local fval = gp_xpos(pnt) - 0.001if fval < 0.0 then

qval = qval + gp_flow(pnt)end_ifpnt = gp_next(pnt)

end_loopqflac = qval /0.15 ; scale for unit thicknessglobal qsol = 1e-10*1e3*10.*(6.*6.-1.2*1.2)/(2.*9.)

enddef plot_seepage_face(plt)

local v1 = vector(c_L,0,1.2)local v2 = vector(c_L,0,1.8)local status = set_line_width(plt,2)staus = draw_line(plt,v1,v2)

endlist @qflac @qsol

save ch2a

FLAC3D Version 4.0


Example 1.9 Steady-state flow through a vertical embankment – Case 2

newtitle "Steady state flow through a vertical embankment - case 2"def setup

global c_perm = 1e-10global c_poro = 0.3global c_kw = 1e3global c_L = 9.global c_h1 = 6.global c_h2 = 1.2

end@setupconfig fluidgen zone brick size 30 1 20 p1 @c_L 0 0 p2 0 0.15 0 p3 0 0 @c_h1; --- fluid flow model ---model fluid fl_isoprop perm @c_perm poros @c_poro ; default value: poros 0.5 biot_c 1set fluid biot offini fmodulus 1e3ini sat 1.0ini pp 6e4 grad 0 0 -1e4fix pp range x -.0001 .0001ini pp 1.2e4 grad 0 0 -1e4 range x 8.99 9.01 z -.1 1.1999ini pp 0 range x 8.99 9.01 z 1.1999 7fix pp range x 8.99 9.01; --- settings ---ini fdensity 1e3ini ftens 0.0set gravity 0 0 -10set mech offset fluid on; --- test ---solve ratio 1e-3

def qflaclocal qval = 0.0local pnt = gp_headloop while pnt # null



end_loop

FLAC3D Version 4.0


qflac = qval /0.15 ; scale for unit thicknessglobal qsol = 1e-10*1e3*10.*(6.*6.-1.2*1.2)/(2.*9.)global qerror = abs(((qval/0.15)-qsol)/qsol)*100.0if qerror > 0.5 then

error = "Outflow error greater than 0.5 ("+string(qerror)+")."end_if


local v1 = vector(c_L,0,1.2)local v2 = vector(c_L,0,1.8)local status = set_line_width(plt,2)staus = draw_line(plt,v1,v2)

endlist @qflac @qsolsave ch2b

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Figure 1.16 Steady-state flow vectors and seepage face solution – Case 1

Figure 1.17 Steady-state flow vectors and seepage face solution – Case 2

FLAC3D Version 4.0


Case 1 is repeated to test fluid flow across two sub-grids that are connected by using either theATTACH command or the INTERFACE command. Example 1.10 lists the commands for flow acrossan ATTACHed grid, and Example 1.11 lists the commands for flow across an interface. Both modelresults are nearly identical to the original Case 1; the values for discharge for both the ATTACH andINTERFACE models are 1.930 × 10−6m2/s. Compare Figures 1.18 and 1.19 to Figure 1.16. Notethat if maxedge is specified with INTERFACE, then the interface will act as an impermeable boundary.Remove the semicolon from the INTERFACE maxedge command to observe this response.

Example 1.10 Steady-state flow through a vertical embankment – Case 1 with ATTACHed grid

newtitle"Steady state flow through a vertical embankment - Case 1 - ATTACHed grid"config fluidgen zone brick size 15 1 10 ...

p0 0.0 0 0 ...p1 4.5 0 0 ...p2 0 0.15 0 ...p3 0 0 6

gen zone brick size 15 2 20 ...p0 4.5 0 0 ...p1 9 0 0 ...p2 4.5 0.15 0 ...p3 4.5 0 6 ...nomerge

attach face range x 4.499 4.501; --- fluid flow model ---model fluid fl_isoprop perm 1e-10 poros 0.3 ; default value: poros 0.5 biot_c 1set fluid biot onini biot_mod 3.333e3 ; could also use: ini fmodulus 1e3ini sat 0.0ini sat 1 range x -.0001 .0001ini sat 1 range x 8.99 9.01 z -.1 1.1999ini pp 6e4 grad 0 0 -1e4 range x -.0001 .0001fix pp range x -.0001 .0001ini pp 1.2e4 grad 0 0 -1e4 range x 8.99 9.01 z -.1 1.1999fix pp range x 8.99 9.01; --- settings ---ini fdensity 1e3ini ftens 0.0set gravity 0 0 -10set mech offset fluid on; --- test ---solve ratio 1.e-3

FLAC3D Version 4.0


def qflaclocal qval = 0.0local pnt = gp_headloop while pnt # null





local v1 = vector(9,0,1.2)local v2 = vector(9,0,1.8)local status = set_line_width(plt,2)staus = draw_line(plt,v1,v2)


save attfl

Figure 1.18 Steady-state flow vectors for ATTACHed grid – Case 1

FLAC3D Version 4.0


Example 1.11 Steady-state flow through a vertical embankment – Case 1 with INTERFACE

newtitle"Steady state flow through a vertical embankment - case 1 - INTERFACE"config fluidgen zone brick size 15 1 10 p0 0.0 0 0 p1 4.5 0 0 p2 0 0.15 0 p3 0 0 6group zone Leftgen zone brick size 15 2 20 p0 5.5 0 0 p1 10 0 0 p2 5.5 0.15 0 p3 5.5 0 6group zone right range group Left not; --- interface ---interface 1 face range x 5.499 5.501interface 1 prop ks 2e9 kn 2e9 fric 15 tension 1e10; interface 1 maxedge 1 : <==== this will make interface impermeable

ini xpos add -1. range group right; --- fluid flow model ---model fluid fl_isoprop perm 1e-10 poros 0.3 ; default value: poros 0.5 biot_c 1set fluid biot onini biot_mod 3.333e3 ; could also use: ini fmodulus 1e3ini sat 0.0ini sat 1 range x -.0001 .0001ini sat 1 range x 8.99 9.01 z -.1 1.1999ini pp 6e4 grad 0 0 -1e4 range x -.0001 .0001fix pp range x -.0001 .0001ini pp 1.2e4 grad 0 0 -1e4 range x 8.99 9.01 z -.1 1.1999fix pp range x 8.99 9.01; --- mechanical model ---model mech elprop bulk 2 shear 1step 0 ; <--- must step 0 with mech on to initialize weighting factors; --- settings ---ini fdensity 1e3ini ftens 0.0set gravity 0 0 -10set mech offset fluid on

; --- test ---solve ratio 1.e-3

def qflaclocal qval = 0.0local pnt = gp_head

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loop while pnt # nulllocal fval = gp_xpos(pnt) - 0.001if fval < 0.0 then




local v1 = vector(9,0,1.2)local v2 = vector(9,0,1.8)local status = set_line_width(plt,2)staus = draw_line(plt,v1,v2)


save intfl

Figure 1.19 Steady-state flow vectors for grid with INTERFACE – Case 1

FLAC3D Version 4.0


1.8.4 Spreading of a Groundwater Mound

This problem studies the transient evolution of a groundwater mound within a porous medium. Themound spreads out and flows along an impervious base under the influence of gravity. It is assumedthat the fluid is incompressible, and that the water is contained initially in a cylindrical region withradius r0 and height h0. The water saturation within the mound is equal to one, and Darcy’s law isapplicable. The mound elevation is compared to an analytic solution as the elevation evolves withtime.

Kochina et al. (1983) have derived the solution for the height, h, of the mound. The solution, asgiven by Barenblatt (1987), assumes a hydrostatic pore-pressure distribution within the mound. Inthe case of zero residual saturation, it may be expressed in the form

h = 1

8√t

[4 − r2

√t]; r ≤ 2 4

√

t (1.110)

where h = h/h0, r = r/r0, t = t/tc, and the characteristic time is given as tc = r20/(2κh0) with

κ = kρwg/(2n) (k is the mobility coefficient, ρw is water density, g is gravity and n is porosity).

This solution applies to long time scales, when the influence of the details of the initial moundgeometry have disappeared.

The results are presented in dimensionless form: the scaled geometrical parameters r0 = 1, h0 =1 are used; and the scaled water properties k = 0.5 ×10−4, ρw = 103 and n = 0.5 are prescribedin the numerical simulation. To model incompressible flow, the bulk modulus of the fluid, Kf ,is given a value that is large compared to the pore-pressure variations in the simulation (Kf = 2×105); the value g = 10 is used for gravity. The FLAC3D grid corresponds to a quarter cylinderand contains 2000 zones (see Figure 1.20). The radius of the model is 2, and its height is 1 unit.The initial saturation is 1 within the mound (radius = 1 unit, height = 1 unit), and zero outside. Theinitial pore-pressure distribution within the mound is hydrostatic. All boundaries are impermeableby default. As time goes on, the mound spreads out under its own weight. The simulation isconducted for a total dimensionless time value of t = 0.85, with intermediate results at t = 0.35,0.45 and 0.65. Example 1.12 lists the commands for this example. The corresponding project file,“gwmound.f3prj,” is located in folder “datafiles\fluid\GroundwaterMound.”

FLAC3D Version 4.0


Figure 1.20 FLAC3D grid and initial state of saturated column

Example 1.12 Spreading of a groundwater mound


config fluid; --- geometry ---gen zone cylinder size 20 10 10 p1 0 2 0 p2 0 0 1 p3 2 0 0; --- fluid flow model ---model fluid fl_isoprop perm 0.5e-4 poros 0.5 biot_c 1set fluid biot onini biot_mod 1e5; if fluid modulus is specified:; set fl biot off; ini fmodulus 0.5e5ini fdensity 1e3ini ftens 0.0; --- initial conditions ---range name mound cyl end1 0 0 -1 end2 0 0 2 rad 1.01ini sat 0.0ini sat 1.0 range nrange mound

FLAC3D Version 4.0


ini pp 1e4 grad 0 0 -1e4 range nrange mound; --- settings ---set gravity 0 0 -10set mech offset fluid onset fluid ratio 0save mound0

; --- test ---solve age 0.35title "Spreading of a groundwater mound (t = 0.35)"save mound1

solve age 0.45title "Spreading of a groundwater mound (t = 0.45)"save mound2

solve age 0.65title "Spreading of a groundwater mound (t = 0.65)"save mound3

solve age 0.85title "Spreading of a groundwater mound (t = 0.85)"save mound4return

Saturation contours are sketched in Figures 1.21 to 1.24. The analytic prediction for the moundheight is calculated by a set of FISH functions, and plotted for comparison in the figures (bold line).The model state at each selected time value is restored, and the file listed in Example 1.13 is calledto create the plots shown in Figures 1.21 to 1.24.

Figure 1.21 corresponds to t = 0.35; at that time, the initial shape of the mound still persists,and a comparison with the analytic solution is probably not yet appropriate. For larger times(see Figures 1.22 to 1.24), the spreading of the groundwater mound described by Eq. (1.110) iscaptured by the numerical solution with reasonable accuracy. The numerical estimate lags behindthe analytical prediction; the discrepancy may be explained by the occurrence of residual saturationin the numerical solution, and by the coarse discretization used in the simulation.

Example 1.13 Spreading of a groundwater mound – saturation and head plots

def moundsetuplocal nval = 100global dx = 2./float(nval)

end@moundsetup;

FLAC3D Version 4.0


def check_errorlocal bt = max(fltime,1.e-20)local xi = 1.0/sqrt(bt)local rmax = 2.0/sqrt(xi)local pnt = gp_headlocal accum_error = 0.0local count = 0loop while pnt # null

local gpe = 0.0local pos = gp_pos(pnt)local rad = mag(vector(xcomp(pos),ycomp(pos),0))if rad < rmax then

local height = xi * (4.0 - rad*rad*xi) * 0.125else

height = 0endiflocal sat = gp_sat(pnt)local zdif = zcomp(pos) - heightif sat > 0.5 then

if zdif > 0 thengpe = zdifcount = count + 1

endifelse

if zdif < 0 thengpe = -zdifcount = count + 1

endifendifaccum_error = accum_error + gpepnt = gp_next(pnt)

end_loopif accum_error > 10 then

local str = ’Accumulated error = ’ + string(accum_error)error = str + ’ with count ’ + string(count)

end_ifend;def plot_mound_elevation(plt)

local bt = max(fltime,1.e-20)local xi = 1./sqrt(bt)local xmax = 2./sqrt(xi)local nloop n (1,100)

local xval = (float(n)-1.)*dxif xval < xmax then

FLAC3D Version 4.0


local zval = xi * (4.-xval*xval*xi) * 0.125else

zval = 0.endiflocal moundi = vector(xval,0,zval)xval = float(n)*dxif xval < xmax then

zval = xi * (4.-xval*xval*xi) * 0.125else

zval = 0.endiflocal moundf = vector(xval,0,zval)local status = set_line_width(plt,2)status = draw_line(plt,moundi,moundf)xval = (float(n)-1.)*dxif xval < xmax then


zval = 0.endifmoundi = vector(0,xval,zval)xval = float(n)*dxif xval < xmax then


zval = 0.endifmoundf = vector(0,xval,zval)status = set_line_width(plt,2)status = draw_line(plt,moundi,moundf)

endloopend;def plot_mound_elevation1(plt)

local bt = max(fltime,1.e-20)local xi = 1./sqrt(bt)local xmax = 2./sqrt(xi)local imax = int(xmax/dx) + 1local nloop n (1,100)

local xval = (float(n)-1.)*dxif xval < xmax then

local zval = xi * (4.-xval*xval*xi) * 0.125else

zval = 0.endif

FLAC3D Version 4.0


local moundi = vector(xval,0,zval)xval = float(n)*dxif xval < xmax then


zval = 0.endiflocal moundf = vector(xval,0,zval)local status = set_line_width(plt,2)status = draw_line(plt,moundi,moundf)

endloopend;def plot_mound_region(plt)

local bt = max(fltime,1.e-20)local xi = 1./sqrt(bt)local xval = sqrt(0.5*xi)local zval = xi * (4.0-xi*xval*xval)/8.0local moundi = vector(xval,0,0)local moundf = vector(xval,0,zval)local status = set_line_width(plt,2)status = draw_line(plt,moundi,moundf)

end;config gpextra 1def c_head

local rhog = 1e4local pnt = gp_headloop while pnt # null

if gp_sat(pnt) > 0.9 thengp_extra(pnt,1) = gp_zpos(pnt)*gp_sat(pnt) + gp_pp(pnt)/rhog

elsegp_extra(pnt,1) = 0.0


end_loopend@c_head;;

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Figure 1.21 Saturation contours and analytical mound elevation at t = 0.35


FLAC3D Version 4.0




FLAC3D Version 4.0


The sketch of flow vectors and head contours in Figure 1.25 corresponds to t = 0.85. (The lackof smoothness in the contour plot is caused by the head jump across the phreatic surface.) Tworegions with water flowing predominantly downward in the core of the mound, and outward in itsperiphery, can be seen in the figure. (The vertical bold line is drawn at the location where the timederivative of the analytical mound elevation vanishes: ∂h/∂t = 0.)

Figure 1.25 Head contours and analytical mound elevation at t = 0.85

1.8.5 One-Dimensional Consolidation

A saturated layer of soil of thickness H = 20 m (shown in Figure 1.26) and large horizontal extentrests on a rigid impervious base. A constant surface load, pz = 105 Pa, is applied on the layerunder undrained conditions. The soil matrix is homogeneous and behaves elastically; the isotropicDarcy’s transport law applies. The applied pressure is initially carried by the fluid but, as time goeson, the fluid drains through the layer surface, transferring the load to the soil matrix. The solutionto this one-dimensional consolidation problem may be expressed in the framework of Biot theory(see Detournay and Cheng 1993).

FLAC3D Version 4.0


z

pz

H

Figure 1.26 One-dimensional consolidation

The diffusion equation for the pore pressure, p, has the form

∂p

∂t− c

∂2p

∂z2= − α

α1S

dσzz

dt(1.111)

where:c = k/S;

S = 1M

+ α2

α1= the storage coefficient;

α1 = K + 43G;

M = is the Biot modulus; andα = is the Biot coefficient.

The boundary conditions are σzz = −pz H(t) and p = 0 at z = H (H(t) is the step function),and uz = 0 and ∂p

∂z= 0 at z = 0.

Because the stress is constant, Eq. (1.111) reduces to

∂p

∂t− c

∂2p

∂z2= 0 (1.112)

with boundary conditions p = 0 at z = H , and ∂p∂z

= 0 at z = 0.

The initial value, p0, for the pore pressure induced from loading of the layer may be derived fromthe fluid constitutive law (p = M(ζ −αεzz)) by considering undrained conditions (ζ = 0) and using

FLAC3D Version 4.0


the one-dimensional mechanical constitutive law (σzz = α1εzz − αp) to express strain in terms ofstresses. It is given by

p0 = α

α1

1

Spz (1.113)

The solution for the pore pressure is

p = 2p0

pz

∞∑

m=0

sin(amz)

ame−a2

mt (1.114)

where:p = ppz

;

am = π2 (2m+ 1);

z = H−zH

; and

t = ct

H 2 .

The vertical displacement, uz, is found by considering the equilibrium equation ∂σzz/∂z = 0,together with the mechanical constitutive equation σzz = α1εzz−αp. By expressing εzz as ∂uz/∂z,we obtain upon integration, taking Eq. (1.114) and the boundary conditions into account,

uz = 2αp0

pz

[ ∞∑

m=0

cos(amz)

a2m

e−a2mt

]

+ y − 1 (1.115)

here uz is defined as uz = α1uzHpz

.

The following properties are prescribed for this example:

dry bulk modulus, K 5 ×108 Pa

dry shear modulus, G 2 ×108 Pa

Biot modulus, M 4 ×109 PaBiot coefficient, α 1.0

permeability, k 10−10 m2

Pa−sec

FLAC3D Version 4.0


The FLAC3D model grid for this problem is a column of 20 zones of unit dimensions lined up alongthe z-axis. The base of the column is fixed, and lateral displacements are restricted in the x- andy-directions. A mechanical pressure, pz, is applied at the top of the column.

At first, flow is prevented and the model is stepped to equilibrium to establish the initial undrainedconditions. At the end of this stage, the stress σzz has the value −pz, in equilibrium with the appliedpressure, and the pore pressure has the initial undrained value p0 = αpz/α1S (see Eq. (1.113)).Drainage is then allowed by setting the pore pressure to zero at the top of the column. The FLAC3D

model is cycled to a flow time of 5000 seconds, which is approximately the magnitude of thecharacteristic time, tc = H 2/c, for this problem.

The analytical solution for the pore pressure, p, and vertical displacement, uz, at the columnmid-height are evaluated using FISH functions, and compared to the numerical solution as timeproceeds. The results are plotted versus fluid-flow time in Figures 1.28 and 1.29. The transfer ofpore pressure to effective stress is illustrated in Figure 1.30, where the evolutions of normalizedtotal stress, −σzz/pz, effective stress, −(σzz + p)/pz, and pore pressure, p/pz, with fluid-flowtime are presented. The FLAC3D data file is listed in Example 1.14. The corresponding project file,“1dConsolidation.f3prj,” is located in folder “datafiles\fluid\1dConsolidation.”

Figure 1.27 FLAC3D grid for one-dimensional consolidation

FLAC3D Version 4.0


Example 1.14 One-dimensional consolidation (coupled)

new projectset fish autocreate off

title "One-dimensional consolidation (coupled)"config fluid

def constantsglobal c_perm = 1e-10global c_biotc = 1.global c_biotm = 4.e9global c_bulk = 5.e8global c_shear = 2.e8local comod = c_bulk + 4. * c_shear / 3.local storage = 1. / c_biotm + c_biotc * c_biotc / comodlocal cv = c_perm / storageglobal hh = 20.global bt = cv / (hh * hh)global pi2 = pi * .5global pz = 1e5global sig0 = - pzglobal p0 = pz * c_biotc / (comod * storage)global uz0 = pz * hh / comod

end@constants

; --- model geometry ---gen zone brick size 1 1 @hhdef point

global pnt = gp_near(0.,0.,10.)global zpnt = z_near(0.5,0.5,10.5)global zz = (hh - gp_zpos(pnt)) / hh

end@point; --- mechanical model ---model mech elset fluid biot onprop bu @c_bulk sh @c_shearfix x yfix z range z -.1 .1ini szz 0.apply szz @sig0 range z 19.9 20.1; --- fluid flow model ---model fluid fl_iso

FLAC3D Version 4.0


prop perm @c_perm biot_c @c_biotcini biot_mod @c_biotmini pp 0; --- fish functions ---def pp10

pp10 = gp_pp(pnt) / pzglobal ft = fltimeglobal c_szz = z_szz(zpnt) / sig0global c_eszz = (z_szz(zpnt) + z_pp(zpnt)) / sig0global c_uz = gp_zdisp(pnt) / uz0

enddef ppsol

global tt = bt * fltimelocal val = 0.0;valz = 0.0local valu = 0.0local mloop m (0,20)

local mm = pi2 * (2. * m + 1.)local c_e = exp(-mm*mm*tt)/mmval = val + sin(mm*zz)*c_evalu = valu + cos(mm*zz)*c_e/mm

end_loopppsol = val*2.*p0/pzglobal zdsol = valu*2.*c_biotc*p0/pz + zz - 1.

end; --- first establish undrained response ---set fluid offsolve ratio 1e-4save cons_und

; --- histories ---hist add fish @pp10hist add fish @ppsolhist add fish @c_szzhist add fish @c_eszzhist add fish @c_uzhist add fish @zdsolhist add fish @tthist add fish @ft

; --- drained response ---fix pp 0 range z 19.9 20.1set fluid onset mech force 0 ratio 1e-4set hist_rep 200

FLAC3D Version 4.0


set fluid substep 1set mech substep 1 auto ;slavesolve age 500set fluid substep 100set mech substep 10 autosolve age 5000save consreturn

Figure 1.28 Comparison between analytical and numerical values of porepressure in a one-dimensional consolidation test

FLAC3D Version 4.0


Figure 1.29 Comparison between analytical and numerical values for verticaldisplacement in a one-dimensional consolidation test

Figure 1.30 Evolution of pore pressure, total and effective stresses in a one-dimensional consolidation test

FLAC3D Version 4.0


Note that, as the stiffness ratio Rk = α2M/(K + 4G/3) increases, the number of zones shouldincrease to keep the error small. This effect is caused by the numerical technique used to evaluatepore-pressure changes caused by volumetric strain in CONFIG fluid. Also, as the ratio Rk increases,the timestep will decrease. However, to keep the error small and the timestep sufficiently largewithout significantly affecting the solution, it is acceptable to limit the value of Biot modulus (orKf ) used in the simulation to a value of about twenty times (K + 4G/3)/α2 (see Section 1.7.1,note 5). To demonstrate the validity of this numerical approach, the analytical solutions for porepressure and displacement corresponding to M = 1.5 × 1010 (Rk � 20) and M = 1.5 × 1015

(Rk � 20 × 105) are compared in Figures 1.31 and 1.32 (data file is Example 1.15). As seen inthose plots, the responses are indeed similar (maximum relative error less than 5%).

Example 1.15 One-dimensional consolidation (analytical solution)

newset fish auto off

title "One-dimensional consolidation (analytical solution)"config fluid

def setupglobal c_Mglobal c_perm = 1e-10global c_biotc = 1.global c_biotm = c_Mglobal c_bulk = 5.e8global c_shear = 2.e8local comod = c_bulk + 4. * c_shear / 3.local storage = 1. / c_biotm + c_biotc * c_biotc / comodlocal cv = c_perm / storageglobal hh = 20.global bt = cv / (hh * hh)global pi2 = pi * .5global pz = 1e5global sig0 = - pzglobal p0 = pz * c_biotc / (comod * storage)global uz0 = pz * hh / comod

enddef ppsol

global ntablocal dt = 100.local iiloop ii (1,50)

local zz = (hh-10.) / hhlocal nt = ntablocal nt1 = ntab + 1local c_fltime = dt * ii

FLAC3D Version 4.0


local tt = bt * c_fltimelocal val = 0.0local valu = 0.0local mloop m (0,20)


end_looplocal valpp = val*2.*p0/pzlocal valzd = valu*2.*c_biotc*p0/pz + zz - 1.xtable(nt,ii) = c_fltimeytable(nt,ii) = valppxtable(nt1,ii) = c_fltimeytable(nt1,ii) = valzd

end_loopendset @c_M=1.5e10 @ntab=1@setup@ppsolset @c_M=1.5e15 @ntab=3@setup@ppsol

def err_pplocal valpp = 0.0local valuz = 0.0local iiloop ii (1,50)

valpp = max(abs(ytable(1,ii)-ytable(3,ii)),valpp)valuz = max(abs(ytable(2,ii)-ytable(4,ii)),valuz)

endlooperr_pp = valppglobal err_uz = valuz

end

table 1 name ’pp M = 1.5e10’table 2 name ’zd M = 1.5e10’table 3 name ’pp M = 1.5e15’table 4 name ’zd M = 1.5e15’

return

FLAC3D Version 4.0


Figure 1.31 Comparison between analytical pore-pressure solutions for twolarge values of M in a one-dimensional consolidation test

Figure 1.32 Comparison between analytical vertical displacement solutionsfor two large values of M in a one-dimensional consolidation test

FLAC3D Version 4.0


In Example 1.16, the numerical simulation is repeated for a Biot modulus of 4 × 1011 Pa (100times larger than in the former numerical calculation), this time using the uncoupling techniquedescribed in Section 1.7.2.3 (the applicability of the uncoupling technique to this problem followsfrom the irrotational character of the solution). In this example, the initial undrained conditionsare obtained using the undrained bulk value for the material and setting Biot modulus to zero. Thepore pressure, which is not updated in this calculation mode, is then initialized at the value p0.The rest of the simulation is carried out in a series of ten time increments to enable a recording ofthe variables history. In each increment, the fluid flow calculation is performed for a time intervalof 500 seconds. During that stage, a scaled value for Biot modulus is used in order to preservethe coupled system true diffusivity (see Eq. (1.103)). Next, Biot modulus is set to zero to preventadditional generation of pore pressure, and the system is run to mechanical equilibrium using thedrained value of the bulk modulus. Fluid and mechanical calculations are repeated until the totalsimulation time reaches 5000 s. The results of this simulation (which does not involve any directcalculation of pore-pressure change due to volumetric straining) are presented in Figures 1.33 to1.35.

Example 1.16 One-dimensional consolidation (uncoupled)

newset fish auto off

title "One-dimensional consolidation (uncoupled)"config fluid

def constantsglobal c_perm = 1e-10global c_biotc = 1.global c_biotm = 4.e9 * 100.global c_bulk = 5.e8global c_shear = 2.e8global comod = c_bulk + 4. * c_shear / 3.local storage = 1. / c_biotm + c_biotc * c_biotc / comodlocal cv = c_perm / storageglobal c_bulku = c_bulk + (c_biotcˆ2) * c_biotmglobal c_biotma= 1. / storageglobal hh = 20.global bt = cv / (hh * hh)global pi2 = pi * .5global pz = 1e5global sig0 = - pzglobal p0 = pz * c_biotc / (comod * storage)global uz0 = pz * hh / comod

end@constants

FLAC3D Version 4.0


; --- model geometry ---gen zone brick size 1 1 @hhdef point

global pnt = gp_near(0.,0.,10.)global zpnt = z_near(0.5,0.5,10.5)global zz = (hh - gp_zpos(pnt)) / hh

end@point

; --- mechanical model ---model mech elset fluid biot onprop bu @c_bulk sh @c_shearfix x yfix z range z -.1 .1ini szz 0.apply szz @sig0 range z 19.9 20.1

; --- fluid flow model ---model fluid fl_isoprop perm @c_perm biot_c @c_biotcini biot_mod @c_biotmini pp 0

; --- fish functions ---def pp10

pp10 = gp_pp(pnt) / pzglobal ft = fltimeglobal c_szz = z_szz(zpnt) / sig0global c_eszz= (z_szz(zpnt) + z_pp(zpnt)) / sig0global c_uz = gp_zdisp(pnt) / uz0

enddef ppsol

local tt = bt * fltimelocal val = 0.0local valu = 0.0local mloop m (0,20)


end_loopppsol = val*2.*p0/pzglobal zdsol = valu*2.*c_biotc*p0/pz + zz - 1.

FLAC3D Version 4.0


end

; --- first establish undrained response ---set fluid off mech onprop bu @c_bulkuini biot_mod 0.0solveini pp @p0save ucons_und

; --- drained response ---fix pp 0 range z 19.9 20.1def my_solve

local iiloop ii (1,10)

global c_age = 500.0*iicommand

prop bu @c_bulkini biot_mod @c_biotmaset fluid on mech offsolve age @c_ageset fluid off mech onini biot_mod 0.0solve ratio 2e-5

end_commandytable(1,ii) = pp10xtable(1,ii) = ftytable(2,ii) = ppsolxtable(2,ii) = ftytable(3,ii) = c_uzxtable(3,ii) = ftytable(4,ii) = zdsolxtable(4,ii) = ftytable(5,ii) = c_szzxtable(5,ii) = ftytable(6,ii) = c_eszzxtable(6,ii) = ft

end_looptable_name(1) = ’Flac Pore Pressure’table_name(2) = ’Analytical Pore Pressure’table_name(3) = ’Flac Vertical Displacement’table_name(4) = ’Analytical Vertical Displacement’table_name(5) = ’Total Stress’table_name(6) = ’Effective Stress’

end@my_solve

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save uconsret

Figure 1.33 Comparison between analytical and numerical values of porepressure in a one-dimensional consolidation test

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Figure 1.34 Comparison between analytical and numerical values for verticaldisplacement in a one-dimensional consolidation test

Figure 1.35 Evolution of pore pressure, total and effective stresses in a one-dimensional consolidation test

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1.8.6 Transient Fluid Flow to a Well in a Shallow Confined Aquifer

A shallow confined aquifer of large horizontal extent is characterized by a uniform initial porepressure, p0, and initial isotropic stress, σ 0

zz. A well, fully penetrating the aquifer, is producingwater at a constant rate, q, per unit depth from time, t = t0. The elastic porous medium ishomogeneous and isotropic, and the flow of groundwater is governed by Darcy’s law. Transienteffects are linked to the compressibility of water and the soil matrix. In this problem, the effect ofpore-pressure changes are small compared to the overburden, and the vertical stress in the aquifermay be assumed to remain constant with time. Also, horizontal strains are neglected compared tothe vertical ones. The conditions of fluid flow to the well are illustrated schematically in Figure 1.36.The numerical solution to this problem is presented using both coupled and uncoupled modelingapproaches.

z

q

Figure 1.36 Flow to a well in a shallow confined aquifer

A cylindrical system of coordinates is chosen with the z-axis pointing upward in the direction ofthe well axis. Substitution of the transport law in the fluid mass-balance equation gives, taking intoconsideration that εrr = εθθ = 0,

∂p

∂t= M(k∇2p − α

∂εzz

∂t) (1.116)

where k is the homogeneous permeability coefficient, M is the Biot modulus and α is the Biotcoefficient.

FLAC3D Version 4.0


Partial differentiation, with respect to time, of the elastic constitutive relationσzz−σ 0zz+α(p−p0) =

α1εzz yields, for constant σzz,

α∂p

∂t= α1

∂εzz

∂t(1.117)

where α1 = K + 4/3G.

Using this last equation to express εzz in terms of p in Eq. (1.116), we obtain, after some manipu-lations,

∂p

∂t= c∇2p (1.118)

where c = k/S is the diffusion coefficient, S = 1/M + α2/α1 is the storage coefficient and, withthe problem being axisymmetric and not dependent on z, the Laplacian of p may be expressed as

∇2p = ∂2p

∂r2+ 1

r

∂p

∂r(1.119)

The solution to this differential equation with boundary conditions

limr→∞p = p0

limr→0

2πr∂p

∂r= q

k

is due to Theis (1935). It has the form

p = − 1

4πE1(u)+ p0 (1.120)

where p = pk/q. The dimensionless variable u is given by

u = r2

4c(t − t0)(1.121)

and E1 is the exponential integral, defined as

FLAC3D Version 4.0


E1(u) =∫ ∞

u

e−ξ

ξdξ (1.122)

The vertical displacement may be obtained by integration of the equilibrium equation ∂σzz/∂z = 0,after expressing σzz in terms of εzz by means of the mechanical constitutive equation and substituting∂εzz/∂z for εzz. This yields, after substitution of the boundary condition, and using Eq. (1.120),

uz = − z

4πE1(u) (1.123)

where uz = ukα1/(αqH) and x = z/H .

The stresses are derived from the mechanical constitutive equations and Eq. (1.120) for p. Theyhave the form

σrr = σθθ = 1

2πE1(u)+ σ 0

zz (1.124)

σzz = σ 0zz

where σ = σkα1/(qαG).

The FLAC3D grid for this problem corresponds to a nine-degree wedge in a hollow cylinder of unitheight. The axis and radius of the well correspond to cylinder axis and radius, respectively. Thecylinder outer radius is selected as 100 m, to model the far boundary of the flow domain. Figure 1.37shows the FLAC3D grid.

FLAC3D Version 4.0


Figure 1.37 FLAC3D grid for a well in a shallow aquifer

A total of 31 zones are used, lined up and graded in the radial direction. The displacements arefixed in the radial and tangential directions, and in the vertical direction at the cylinder base. Avertical pressure of magnitude −σ 0

zz is applied at the top of the model.

The properties for this example are defined:

dry bulk modulus, K 118 MPadry shear modulus, G 71 MPawater bulk modulus, Kf 2 GPaBiot coefficient, α 1.0porosity, n 0.4

permeability, k 2.98 ×10−8 m2

Pa−sec

The initial pore pressure is 147 kPa, and the initial isotropic stress is −147 kPa. Because theBiot coefficient is equal to one (incompressible soil grains), the Biot modulus is equal to the ratiobetween water bulk modulus and porosity (in this case, M = 5 GPa). The well-pumping rate perunit aquifer thickness is 2.21 10−3 m2/s, and the well radius, rw, is selected as 1 m.

Stresses and pore pressures are initialized to the values given above. The well flow-rate is modeledas a surface flux of magnitude q/(2πrw) applied to the well radius r = rw.

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Both coupled and uncoupled solutions are provided for this problem to a time of 32 seconds, withintermediate results at 4, 8 and 16 seconds. Both solutions are included in the FLAC3D data filelisted in Example 1.17. The corresponding project file, “WellConfAquifer.f3prj,” is located in folder“datafiles\fluid\WellConfAquifer.”

The coupled problem is solved (SET fluid on mech on) using the explicit solution algorithm. Themaximum out-of-balance mechanical force is limited to 10.0, the maximum number of mechanicalsub-steps in the coupled fluid-mechanical calculation step is limited to 1000, and the mechanicalprocess is the “slave” module to the master fluid-flow process. This is accomplished with thecommands

set mech substep 1000 autoset mech force 10

By specifying these commands, the out-of-balance mechanical force will be kept to a small valuewhile the fluid-flow calculation proceeds.

This example is pore-pressure driven, and the value for the stiffness ratio, Rk , is approximately23 for the specified fluid bulk modulus of 2 GPa. Thus, the flow calculation may be uncoupledfrom the mechanical calculation, and the approach discussed in Section 1.7.2.3 may be applied.The fluid modulus during the flow-only step is defined by Eq. (1.103) in order to preserve thediffusivity of the system. During the mechanical-only step, the fluid modulus is set to zero toprevent further adjustments by volumetric strains. The following commands are applied for theuncoupled calculation for a 4-second flow time:

set mech force 10set fluid on mech offini fmod = uwbsolve age 4.set fluid off mech onini fmod = 0.0solve

The FISH variable uwb is the adjusted fluid modulus calculated by Eq. (1.103). Note that, ifconditions are such that Rk <<< 1, it is not necessary to adjust the fluid modulus during the flowcalculation because the diffusivity will be accurate.

The analytical solutions for pore pressure, stresses and vertical displacement are programmed asFISH functions in Example 1.17. The exponential integral function used in the analytical solutionsis programmed as a separate FISH function contained in file “exp-int.fis” (see Example 1.18).Analytical and numerical values are stored in tables. The results are then compared in graphicalform. The pore-pressure comparison at selected times is presented in Figure 1.38 for the coupledsolution, and in Figure 1.39 for the uncoupled solution. The stresses and vertical displacementvalues at 32 seconds are processed by the FISH function well 32, and illustrated in Figures 1.40and 1.42 for the coupled solution, and in Figures 1.41 and 1.43 for the uncoupled solution.

The results for both the coupled and uncoupled solutions are essentially identical, and comparewell with the analytical solution. The uncoupled solution is reached much more quickly than the

FLAC3D Version 4.0


coupled solution. Note that the coupled calculation requires more than 500,000 steps, while theuncoupled calculation requires approximately 16,000 steps.

Example 1.17 Transient fluid flow to a well in a shallow aquifer


titleTransient fluid flow to a well in a shallow confined aquifer

config fluid

define constants ; model constantsglobal n1 = 31 ; zones in x-directionglobal n2 = 1 ; zones in y-directionglobal w_bulk = 2e9 ; water bulk modulusglobal c_poro = 0.4 ; porosityglobal c_biom = w_bulk / c_poro ; Biot modulusglobal c_bioc = 1. ; Biot coefficientglobal eps = 1.e-3 ; tolerance for well radiusglobal rn = 1. ; well radiusglobal qtot = 2.21e-3 ; well pumping rate /

; unit aquifer thicknessglobal qin = -qtot/(2.0*pi*rn) ; equivalent fluxglobal rin = rn + epsglobal p0 = 147000. ; initial pore pressureglobal c_kw = 2.98e-8 ; Flac permeabilityglobal c_e = qtot / (4. * pi * c_kw * p0)global c_k = 11.8e7 ; bulk modulusglobal c_g = 7.1e7 ; shear modulusglobal sig0 = -1.47e5 ; initial isotropic stressglobal al1 = c_k + 4. * c_g / 3.global ss = (1. / c_biom + c_biocˆ2 / al1) * 0.25 / c_kw ; 1./4cglobal stor = c_poro/w_bulk + 1.0/al1 ; storativityglobal uwb = c_poro / stor ; adjusted fluid modulusglobal R_k = w_bulk / (c_poro * al1) ; stiffness ratioglobal err = get_array(15)

end@constants

; --- model geometry (hollow cylinder - 9 degree wedge) ---gen zone cshell p0 0. 0 1. p1 1.e2 0 1. &

p2 0. 0 0. p4 1.e2 0 0. &p3 0.98769e2 0.15643e2 1. &p5 0.98769e2 0.15643e2 0. &

FLAC3D Version 4.0


dim 1 1 1 1 size @n1 @n2 1 ratio 1.1 1 1range name in cyl end1 0 0 -100 end2 0 0 100 rad @rinrange name out cyl end1 0 0 -100 end2 0 0 100 rad 99. notrange name xl y 0 z 0; --- mechanical model ---model mech elprop bu @c_k sh @c_gfix x yfix z range z -0.001 0.001ini sxx @sig0ini syy @sig0ini szz @sig0apply szz @sig0 range z 1; --- fluid flow model ---model fluid fl_isoprop perm @c_kw poro @c_poroini fmod @w_bulkini pp @p0apply discharge @qin range nrange in; --- fish functions ---call exp-int.fis suppress

define sol_pp(tab1,tab2)local pnt = gp_headlocal count = 0local avg = 0err(tab1) = 0loop while pnt # null

local rad = sqrt(gp_ypos(pnt)ˆ2 + gp_zpos(pnt)ˆ2)if rad < 1.e-4 then

local x = gp_xpos(pnt)if x > 0.99 then

count = count + 1table(tab1,x) = gp_pp(pnt) / p0local ft = fltimelocal e_val = ss * x * x / ftlocal val = exp_int(e_val)table(tab2,x) = - val * c_e + 1.err(tab1) = err(tab1) + (table(tab2,x)-table(tab1,x))ˆ2avg = avg + table(tab2,x)


end_loopavg = abs(avg)err(tab1) = sqrt(count*err(tab1)) / avg

FLAC3D Version 4.0


end; --- setting ---set fluid on

save ex1_17-ini;; --- pumping (coupled analysis) ---set fluid substep 100set mech substep 1000 auto ;slaveset mech force 10.solve age 4.@sol_pp(1,2)solve age 8.@sol_pp(3,4)solve age 16.@sol_pp(5,6)solve age 32.@sol_pp(7,8)

save ex1_17-cpl

; --- pumping (uncoupled analysis) ---restore ex1_17-iniset mech force 10set fluid on mech offini fmod = @uwbsolve age 4.set fluid off mech onini fmod = 0.0solve@sol_pp(1,2);set fluid on mech offini fmod = @uwbsolve age 8.set fluid off mech onini fmod = 0.0solve@sol_pp(3,4);set fluid on mech offini fmod = @uwbsolve age 16.set fluid off mech onini fmod = 0.0solve

FLAC3D Version 4.0


@sol_pp(5,6);set fluid on mech offini fmod = @uwbsolve age 32.set fluid off mech onini fmod = 0.0solve@sol_pp(7,8)save ex1_17-ucpl;return

Example 1.18 Exponential integral function – “EXP-INT.FIS”

; --- Exponential integral E1(e_val) ---; Input: e_val;define exp_int(e_val)

local e_e1if e_val < 0.0 then

local ii = out(’ Argument of Exponential function must be positive’)exit

end_ifif e_val = 0.0 then

exp_int = 1.e12exit

end_ifif e_val < 1. then

e_e1 = ((.00107857 * e_val - .00976004) * e_val + .05519968) * e_vale_e1 = ((e_e1 - .24991055) * e_val + .99999193) * e_valexp_int = e_e1 - .57721566 - ln(e_val)

elsee_e1 = .250621 + e_val * (2.334733 + e_val)e_e1 = e_e1 / (1.681534 + e_val * (3.330657 + e_val))exp_int = e_e1 * exp(-e_val) / e_val

end_ifendreturn

FLAC3D Version 4.0


Figure 1.38 Pore-pressure distribution at 4, 8, 16 and 32 seconds– coupled solution(analytical values = lines; numerical values = symbols)

Figure 1.39 Pore-pressure distribution at 4, 8, 16 and 32 seconds– uncoupled solution(analytical values = lines; numerical values = symbols)

FLAC3D Version 4.0


Figure 1.40 Radial, tangential and vertical stress distributions at 32 seconds– coupled solution(analytical values = lines; numerical values = symbols)

Figure 1.41 Radial, tangential and vertical stress distributions at 32 seconds– uncoupled solution(analytical values = lines; numerical values = symbols)

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Figure 1.42 Radial displacement distribution at 32 seconds– coupled solution(analytical values = line; numerical values = symbols)

Figure 1.43 Radial displacement distribution at 32 seconds– uncoupled solution(analytical values = line; numerical values = symbols)

FLAC3D Version 4.0


1.8.7 Pressuremeter Test

The pressuremeter test is used to determine in-situ mechanical properties of soils (Wood 1990).A long, rubber membrane is expanded against the walls of a vertical borehole (Figure 1.44). Thepressure inside the membrane is constant. Radial displacements of the borehole wall are measuredas a function of the pressure. The soil deforms in plane strain in the plane normal to the boreholeand sufficiently distant from the ends of the membrane.

Figure 1.44 Cylindrical cavity expansion in pressuremeter test

The borehole (radius a = 0.03 m) is drilled in homogeneous, isotropic soil. The soil is assumedto behave as a linearly elastic material saturated with groundwater. The following mechanicalproperties of the soil and groundwater are assumed in this problem:

shear modulus (G) 11.1 MPabulk modulus (K) 33.3 MPaporosity (n) 0.48

soil permeability (k) 1.02 ×10−14 m2

Pa−secbulk modulus of water (Kf ) 500 MPa

The initial state of (total) stress in the soil is isotropic: σ1 = σ2 = σ3 = −327.87 kPa, while theinitial pore pressure is p = pi = 147.0 kPa.

FLAC3D Version 4.0


The soil is allowed to consolidate for 300 seconds after the drilling of the borehole. After theconsolidation, the pressure inside the rubber membrane is increased from zero to 1 MPa during 600seconds. (The rubber membrane prevents further drainage of the groundwater into the borehole.)

The analytical solution for two-dimensional consolidation of a borehole in an elastic medium wasobtained by Detournay and Cheng (1988). The drilling of the borehole is simulated by removingthe stress acting on the inner boundary of the borehole and setting the pore pressure to zero at timet = 0. Since the initial stress is isotropic, the loading conditions can be decomposed into two modes:(1) mode 1, an isotropic stress; and (2) mode 2, an initial pore-pressure distribution. The boundaryconditions at the wall of the borehole for each loading mode can be expressed as follows:

(1) mode 1

σrr = −σiσrθ = 0

p = 0

(2) mode 2

σrr = 0

σrθ = 0

p = −pi

The stresses and displacements due to mode 1 loading are described by the classical Lamé solution.Since the volumetric strain computed from the Lamé solution is zero throughout the domain, themode 1 loading does not generate pore pressure, and deformation takes place instantaneously. Theevolution of the pore-pressure field due to mode 2 loading is governed by a homogeneous diffusionequation. The deformation and stress fields can be calculated from the pore-pressure field. Theproblem is solved in the Laplace transform domain, and the solutions are transformed back to thetime domain using the numerical inversion method developed by Stehfest (1970). The completesolution is described by Detournay and Cheng (1988). The analytical solutions are calculated andimported into FLAC3D tables for comparison to the numerical results.

Because the problem is axisymmetric, it is simulated using a row of 61 zones forming a truncatedwedge of 9◦ angle. The geometry of the model and boundary conditions are illustrated in Figure 1.45.The FLAC3D grid is shown in Figure 1.46. The far-field boundary is at radius b = 1.50 m and canbe considered at infinity if radius b is scaled to the radius of the borehole (a = 0.03 m). (The lengthresulting from the diffusivity of the model, and time of the simulation, are also much smaller thanb.)

FLAC3D Version 4.0


Figure 1.45 Domain of FLAC3D simulation

Figure 1.46 FLAC3D grid

FLAC3D Version 4.0


The simulation is conducted in the steps corresponding to the actual operations in the field test:

(1) The total stress on the contour of the borehole is reduced to zero, simulating drillingof the borehole. Although the total stress (in the FLAC3D model) is reduced to zero insteps, the change in real flow time is instantaneous (i.e., the model undergoes undraineddeformation). (The fluid flow calculation is turned off: SET fluid off.) The model isiterated to reach mechanical equilibrium.

(2) The pressure boundary condition at the contour of the borehole is set to zero. The modelconsolidates for 300 seconds (SET fluid on), resulting in the drainage of the groundwaterinto the borehole.

(3) The contour of the borehole is defined as impervious (due to installation of the rubbermembrane), and the pressure boundary condition is applied on the contour of the borehole:1.0 MPa in 100 increments at each 6 seconds. That is, the soil consolidates under theapplied load for 6 seconds before the next load increment (0.01 MPa) is applied.

The FLAC3D model requires approximately 9 MB RAM and takes less than 15 minutes to run allstages of the simulation on a 2.8 GHz Pentium IV computer.

The profiles of the normalized pore pressure, p/pi , normalized radial stress, σrr/σi , and tangentialstress, σθθ/σi (as a function of the normalized radius r/a), after 300 seconds of consolidation,calculated from FLAC3D and using the closed-form solution from Detournay and Cheng (1988),are shown in Figures 1.47 and 1.48. Agreement between the curves is very good.

The profiles of the normalized pore pressures, and the radial and tangential stresses after 600 secondsof pressurization of the borehole are shown in Figures 1.49 and 1.50.

FLAC3D Version 4.0


Figure 1.47 Pore-pressure profiles – 300 seconds consolidation

Figure 1.48 Profiles of radial and tangential normal stresses – 300 secondsconsolidation

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Figure 1.49 Pore-pressure profiles – 600 seconds consolidation

Figure 1.50 Profiles of radial and tangential normal stresses – 600 secondsconsolidation

FLAC3D Version 4.0


The pressure variations in the pressuremeter test are often such that nonlinear, plastic deformationsare induced in a soil. Therefore, the same problem is simulated using a Mohr-Coulomb model forplastic deformation of the soil. The following strength parameters are assumed in the simulation:

friction angle (φ) 22◦dilation angle (ψ) 10◦cohesion (c) 26 kN/m2

The profiles of the normalized pore pressures, and the normalized radial and tangential stressesafter 600 seconds of pressurization of the borehole in a Mohr-Coulomb material, are shown inFigures 1.51 and 1.52.

The numerical solution for a linearly elastic material is generated using the data file listed in Ex-ample 1.19. The input data file for a Mohr-Coulomb material is the same, except that: (1) thematerial model is declared a Mohr-Coulomb material (MODEL mohr); and (2) the correspond-ing material properties are added (PROP fric 30 dil 10 coh 26000). The data file “pressuremeter-maket.f3fis” generates the tables with the profiles of the normalized pore pressure and the normal-ized stresses (see Example 1.20). The data file “preana.dat” contains tables in which the analyticalelastic solutions for pore pressure, radial and tangential stress profiles, after 300 seconds of pres-surization, are stored. The corresponding project file, “pressuremeter.f3prj,” is located in folder“datafiles\fluid\pressuremeter.”

Figure 1.51 Pore-pressure profile – 600 seconds consolidation(Mohr-Coulomb material)

FLAC3D Version 4.0


Figure 1.52 Profiles of radial and tangential normal stresses – 600 secondsconsolidation (Mohr-Coulomb material)

Example 1.19 Pressuremeter test

;--------------------------------------------------------------------; pressuremeter test in poro-elastic material; (a) Elastic model;--------------------------------------------------------------------new projectset fish autocreate off

title ’Pressuremeter - elastic model’

call pressuremeter-maket.f3fis suppress

def test0global tim0=clock

enddef test1

global tim=(clock-tim0)/100.0end@test0

config fluid

FLAC3D Version 4.0


; --- constants ---define gcons

global n1 = 61global w_bulk = 5e8global c_poro = 0.48global c_biom = w_bulk / c_poro ;Biot modulus

end@gcons; --- locations of history points ---define findid

local c_idlocal pnt = gp_headloop while pnt # null

c_id = gp_id(pnt)if c_id = 1 then

global id1 = pntend_ifif c_id = 17 then





global id6 = pntend_ifpnt = gp_next(pnt)

end_looppnt = zone_headloop while pnt # null

c_id = z_id(pnt)if c_id = 1 then

global zid1 = pntend_ifif c_id = 5 then

global zid2 = pntend_ifif c_id = 10 then

global zid3 = pntend_if

FLAC3D Version 4.0


if c_id = 15 thenglobal zid4 = pnt

end_ifif c_id = 20 then

global zid5 = pntend_ifif c_id = 31

global zid6 = pntend_ifpnt = z_next(pnt)

end_loopend

define radinradin = 1.001*gp_xpos(id1)

endgen zone brick p0 0.03 0 0 p1 1.5 0 0 &

p2 0.03 0 -0.01 p4 1.5 0 -0.01 &p3 2.96307e-2 0.46930e-2 0 p6 1.48153 0.23465 0 &p5 2.96307e-2 0.46930e-2 -0.01 p7 1.48153 0.23465 -0.01 &size @n1 1 1 ratio 1.1 1 1

range name out cyl end1 0 0 -10 end2 0 0 10 rad 1.48 notrange name xl y -.001 .001 z -.001 .001@findid; --- mechanical model ---; elasticmodel mech elasticprop bu 3.33e7 sh 1.11e7ini sxx -327870ini syy -327870ini szz -327870apply nstress -327870 range nrange outapply nstress -327870 range cyl end1 0 0 -10 end2 0 0 10 rad @radin; --- gw model ---model fluid fl_isoset fluid biot onprop perm 1.02e-14ini biot_mod @c_biomini pp 147000; --- histories ---def gwtime

gwtime = fltimeglobal rad = gp_xpos(id1)global pp1 = gp_pp(id1)global pp2 = gp_pp(id2)global pp3 = gp_pp(id3)

FLAC3D Version 4.0


global pp4 = gp_pp(id4)global pp5 = gp_pp(id5)global pp6 = gp_pp(id6)global sx1 = z_sxx(zid1)global sx2 = z_sxx(zid2)global sx3 = z_sxx(zid3)global sx4 = z_sxx(zid4)global sx5 = z_sxx(zid5)global sx6 = z_sxx(zid6)

endset hist_rep 50his add fish @gwtimehis add fish @radhis add fish @pp1his add fish @pp2his add fish @pp3his add fish @pp4his add fish @pp5his add fish @pp6his add fish @sx1his add fish @sx2his add fish @sx3his add fish @sx4his add fish @sx5his add fish @sx6; --- excavate: reduction of total pressure to zero in steps ---define slow_excav

local nnloop nn (1,100)

local fac = 1.0 - float(nn) / 100.0global ap_press = -327870.0 * faclocal ii = out(’ applied pres = ’+string(ap_press))radincommand

apply nstr @ap_press range cyl end1 0 0 -10 end2 0 0 10 rad @radinsolve force 0.1

end_commandend_loop

end@radinfix pp range cyl end1 0 0 -10 end2 0 0 10 rad @radin

fix y z range y -.00001 .00001apply nvel 0. pl nor -0.46930e-2 2.96307e-2 0 &

dvel 0. pl nor -0.46930e-2 2.96307e-2 0 &range plane nor -0.46930e-2 2.96307e-2 0 orig 0 0 0 dist 0.00001

FLAC3D Version 4.0


set fluid offset largeset echo off@slow_excavset echo onsolve for 0.001save pre1; --- fish function ---define charge

local zzlocal pas = 0global pisglobal abcloop zz (0,100)

abc = -1e4 * zzpas = pas + 6pis = 300 + paslocal ii = out(’ applied mec. pressure = ’+string(-abc))radincommand

apply nstress @abc range cyl end1 0 0 -10 end2 0 0 10 rad @radinset mech substep 1set fluid offsolve for 0.001set fluid onset mech substep 100000 auto ;slaveset mech force 0.01solve age @pis

end_commandend_loop

end; --- let water flow out for 300 s ---set fluid on@radinini pp 0 range cyl end1 0 0 -10 end2 0 0 10 rad @radinset mech substep 100000 auto ;slaveset mech force 0.01solve age 300save pre2

; --- apply pressure inside the borehole ---@radinfree pp range cyl end1 0 0 -10 end2 0 0 10 rad @radin@charge@test1list @tim

FLAC3D Version 4.0


save pre3return

Example 1.20 Generate tables

; numerical solution for pressuremeter test; table 1: pore pressure profile; table 2: radial stress profile; table 3: tangential stress profile; table 4: vertical stress profiledefine maket

local sig_0 = -327870local pp_0 = 147000local a = 0.03local ang = 4.5*degradlocal sagn = sin(ang)local cagn = cos(ang)local ss = sagn*sagnlocal cc = cagn*cagnlocal s2a = 2.*sagn*cagnlocal tab1 = 1local tab2 = 2local tab3 = 3local tab4 = 4local pnt = zone_headloop while pnt # null

local x = z_xcen(pnt)local y = z_ycen(pnt)local z = z_zcen(pnt)local rad = sqrt(x*x+y*y+z*z)/atable(tab1,rad) = z_pp(pnt)/pp_0table(tab2,rad) = (z_sxx(pnt)*cc+z_sxy(pnt)*s2a+z_syy(pnt)*ss)/sig_0table(tab3,rad) = (z_sxx(pnt)*ss-z_sxy(pnt)*s2a+z_syy(pnt)*cc)/sig_0table(tab4,rad) = z_szz(pnt)/sig_0pnt = z_next(pnt)

end_loopend

; Compare tables tab1 and tab2define checkError(tab1,tab2,tol)

local str = ’mismatch between tables ’ + string(tab1)str = str + ’ and ’ + string(tab2)

local ielocal tab = tab1if table_size(tab1) > table_size(tab2) then

FLAC3D Version 4.0


tab1 = tab2tab2 = tab

endiflocal err = 0local max_ = -1e20loop ie (1,table_size(tab1))

local xval1 = xtable(tab1,ie)local yval1 = ytable(tab1,ie)local yval2 = table(tab2,xval1)max_ = max(max_,yval1)err = err + (yval2-yval1)ˆ2

endlooperr = sqrt(err/table_size(1)) / max_if err > tol then

error = str + ’ error is : ’ + string(err)endif

end

return

FLAC3D Version 4.0


1.9 Verification of Concepts, and Modeling Techniques for Specific Applications

1.9.1 Solid Weight, Buoyancy and Seepage Forces

When fluid flows through a porous medium there are, following Terzaghi (1943) and Taylor (1948),three forces acting per unit volume on the solid matrix: the solid weight, the buoyancy, and thedrag or seepage force (also see Bear 1972). These forces are automatically taken into account inthe FLAC3D formulation. This may be shown as follows.

In FLAC3D, equilibrium is expressed using total stress:

∂σij

∂xj+ ρsgi = 0 (1.125)

where ρs is the undrained (saturated bulk) density, and gi is gravitational vector. (Note that Einsteinnotation convention for summation over repeated indices applies to the equations in this section).Undrained density may be expressed in terms of drained density, ρd , and fluid density, ρw, usingthe expression

ρs = ρd + nsρw (1.126)

where n is porosity, and s is saturation. The definition of effective stress is

σij = σ ′ij − pδij (1.127)

Substitution of Eqs. (1.126) and (1.127) in Eq. (1.125) gives, after some manipulations,

∂σ ′ij

∂xj+ ρdgi − (1 − n)

∂p

∂xi− nγw

∂φ

∂xi= 0 (1.128)

where we have introduced fluid unit weight γw and piezometric head φ as

γw = ρwg (1.129)

φ = p

ρwg− xkgk

g(1.130)

and g is the gravitational magnitude.

In Eq. (1.128), the term ρdgi can be associated with solid weight, (1 − n)∂p∂xi

with buoyancy, and

nγw∂φ∂xi

with seepage force (drag).

FLAC3D Version 4.0


1.9.1.1 A Simple Example Illustrating Solid Weight, Buoyancy and Seepage Forces

A simple example is given below to illustrate the contribution of these individual terms in thecontext of FLAC3D methodology. We consider a layer of soil of large lateral extent and thickness,H = 10 m, resting on a rigid base. The layer is elastic, the drained bulk modulus, K , is 100 MPa,and the shear modulus, G, is 30 MPa. The density of the dry soil, ρd , is 500 kg/m3. The porosity,n, is uniform with a value of 0.5. The mobility coefficient, k, is 10−8 m2/(Pa-sec). The fluid bulkmodulus, Kw, is 2 GPa, and gravity is set to 10 m/sec2.

Initially, the water table is at the bottom of the layer, and the layer is in equilibrium under gravity.We study the heave of the layer when the water level is raised, and also the heave or settlementunder a vertical head gradient.

The problem is one dimensional; the FLAC3D model is a mesh composed of a single column of 10zones in the z-direction. The axes origin is at the bottom of the model. The mechanical boundaryconditions correspond to roller boundaries at the base and sides of the model. The fluid-flowboundary conditions are described for the individual cases below.

This example is run using the groundwater configuration (CONFIG fluid). The coupled groundwater-mechanical calculations are performed using the basic fluid-flow scheme. The calculation timesare quite short for this small model; for larger models, uncoupled modeling can be applied to speedthe calculation.

For reference, in comparison of FLAC3D results to the analytical solutions, the one-dimensionalincremental stress-strain relation for this problem condition is

σzz + α p = (K + 4G/3) εzz (1.131)

where α is the Biot coefficient (set equal to 1 for this simulation), K is the drained bulk modulus,G is the shear modulus, and εzz is the vertical strain.

Solid Weight – We first consider equilibrium of the dry layer. The dry density of the material isassigned, and the saturation is initialized to zero (the default value for saturation is 1 in CONFIGfluid mode). The flow calculation is turned off, and the mechanical calculation is on. The valueof fluid bulk modulus is set to zero to prevent any generation of pore pressure under volumetricstraining for this stage. The model is cycled to equilibrium. By integration of Eq. (1.125) appliedto the dry medium, we obtain

σ (1)zz = −ρdg(H − z) (1.132)

Vertical stress at the end of the FLAC3D simulation is plotted versus elevation in Figure 1.53. Thevalues match those obtained for equilibrium under gravity of the dry medium (Eq. (1.132)), asexpected.

FLAC3D Version 4.0


Figure 1.53 Vertical stress versus elevation – dry layer

The vertical displacement at the top of the model is found from the equation

u = − ρdgH2

2(K + 4G/3)(1.133)

The calculated value from FLAC3D matches the analytical value at this stage (−1.786 ×10−3 m).Note that the equilibrium ratio limit (SET mech ratio) is reduced to 10−6 to provide this level ofaccuracy for this example.

Buoyancy – We continue this example by raising the water table to the top of the model. We resetthe displacements to zero, and assign the fluid properties listed above. The pore pressure is fixedat zero at the top of the model, and the saturation is initialized to 1 throughout the grid. (Note thata fluid-flow calculation to steady state is faster if the state starts from an initial saturation 1 insteadof a zero saturation.) Fluid-flow and mechanical modes are both on for this calculation stage, anda coupled calculation is performed to reach steady state. The code uses the saturated density forthis calculation, as determined (internally) from Eq. (1.126). By integration of Eq. (1.125) for thesaturated medium, we obtain

σ (2)zz = −ρsg(H − z) (1.134)

FLAC3D Version 4.0


The comparison of total vertical stress profile from FLAC3D to that from Eq. (1.134) is shown inFigure 1.54:

Figure 1.54 Vertical stress versus elevation – saturated layer

At steady state, the pore pressure is hydrostatic:

p(2) = −ρwg(H − z) (1.135)

Contours of pore pressure at steady state are shown in Figure 1.55.

FLAC3D Version 4.0


Figure 1.55 Pore pressure contours at steady state – saturated layer

The vertical displacement induced by raising the water table is now upwards. The amount of heaveis calculated starting from Eq. (1.131). We write this equation in the form

(σ (2)zz − σ (1)zz )+ (p(2) − p(1)) = (K + 4G/3)du

dz(1.136)

For this example, p(1) = 0. After substitution of Eqs. (1.132), (1.134) and (1.135) into Eq. (1.136),we obtain, after some manipulation,

−[

(ρs − ρw)− ρd

]

g(H − z) = (K + 4G/3)du

dz(1.137)

The term (ρs − ρw) is the buoyant density. Substitution of Eq. (1.126) for undrained density inEq. (1.137) produces

(1 − n)ρwg(H − z) = (K + 4G/3)du

dz(1.138)

Finally, after integration between 0 and H, we obtain

FLAC3D Version 4.0


u = (1 − n)ρwg

(K + 4G/3)

[

H − z

2

]

z (1.139)

and, for z = H , this gives

u = − (n− 1)ρwgH 2

2(K + 4G/3)(1.140)

The final numerical displacement at the top of the model compares well with the analytical value(+1.786 ×10−3 m). The displacements, induced upward, are plotted in Figure 1.56:

Figure 1.56 Heave of the layer at steady state – saturated layer

Additional Rise in Water Table – We continue from this stage, and model the effect of an additionalrise in the water level on the layer. This time, the water table is raised to 20 m above the top of themodel. The corresponding hydrostatic pressure is p = ρwgh, where h is 20 m and p = 0.2 MPa.

We reset displacements to zero and apply a pressure of 0.2 MPa at the top of the model. A fluidpore pressure is applied (with FIX pp 2e5), as is a mechanical pressure (with APPLY nstress), alongthe top boundary. We now perform the coupled calculation again for an additional 500 secondsof fluid-flow time. No further movement of the model is calculated. This is because the absoluteincrease in σzz is balanced by the increase in pore pressure, and the Biot coefficient is set to 1. Thus,no displacement is produced. At the end of this stage, the hydrostatic pore pressure is given by

FLAC3D Version 4.0


p(3) = p(3)b

[

1 − z

H

]

+ ptz

H(1.141)

where p(3)b is the fluid pressure at the base of the layer, and pt is the pressure at the top. For this

case, p(3)b = 0.3 MPa, and pt = 0.2 MPa.

Seepage Force (Upwards Flow) – We now study the scenario in which the base of the layer is incontact with a high-permeability over-pressured aquifer. The pressure in the aquifer is 0.5 MPa.We continue from the previous stage, reset displacements to zero, and apply a pore pressure of 0.5MPa at the base (FIX pp 5e5). The coupled mechanical-flow calculation is performed until steadystate is reached. The plot of displacement vectors at this stage, shown in Figure 1.57, indicatesheave as a result of the upwards flow.

Figure 1.57 Heave of the layer at steady state – seepage force from over-pressured aquifer

The analytical solution for the heave can be calculated from Eq. (1.131). There is no change intotal stress, and so the term σzz drops out. Also, the Biot coefficient is equal to unity. Thus wecan write

p(4) − p(3) = (K + 4G/3)du

dz(1.142)

FLAC3D Version 4.0


where p(3) is given by Eq. (1.141), and p(4) is the steady-state pore pressure distribution at the endof this stage. This pore pressure is given by

p(4) = p(4)b

[

1 − z

H

]

+ ptz

H(1.143)

where p(4)b = 0.5 MPa. Substitution of Eqs. (1.143) and (1.141) into Eq. (1.142), and furtherintegration produces

u = p(4)b − p

(3)b

(K + 4G/3)z[

1 − z

2H

]

(1.144)

For y = H , we obtain

u = p(4)b − p

(3)b

(K + 4G/3)

H

2(1.145)

The FLAC3D result for surface heave compares directly to this result (u = 7.143 ×10−3 m).

Seepage Force (Downwards Flow) – The seepage force case is repeated for the scenario in whichthe base of the layer is in contact with a high-permeability under-pressured aquifer. This time, apressure value ofp(5)b = 0.1 MPa is specified at the base. The displacements are reset and the coupledcalculation is made. The layer settles in this case, which can be seen from the displacement vectorplot in Figure 1.58. The analytical value for the displacement may be derived from Eq. (1.145)after replacing p(5)b for p(4)b . The FLAC3D settlement compares well with the analytical settlementof u = −7.143 ×10−3 m.

FLAC3D Version 4.0


Figure 1.58 Settlement of the layer at steady state – seepage force from under-pressured aquifer

The complete data file for all of these cases is listed below in Example 1.21. The correspondingproject file, “WeightBuoyancySeepage.f3prj,” is located in folder “datafiles\fluid\WeightBuoyan-cySeepage.”

Example 1.21 Solid weight, buoyancy and seepage forces

;; Solid weight, buoyancy and seepage forces


title ’Buoyancy - fully coupled’

config fluid

define setupglobal m_bu = 1e8 ; drained bulk modulusglobal m_sh = 0.3e8 ; shear modulusglobal m_d = 500.0 ; material dry mass densityglobal m_n = 0.5 ; porosityglobal w_d = 1000.0 ; water mass densityglobal _grav = 10.0 ; gravity

FLAC3D Version 4.0


global _H = 10.0 ; height of layer;

global mgrav = -_gravend@setup

define num_dislocal al1 = m_bu+4.0*m_sh/3.0global m_rho = m_d+m_n*w_d ; material wet densityglobal ana_dis1 = -(m_d)*_grav*_Hˆ2/(2.0*al1)global ana_dis2 = -(m_n-1.0)*w_d*_grav*_Hˆ2/(2.0*al1)global ana_dis3 = 0.0global ana_dis4 = 2e5*_H/(2.0*al1) ; new pp - old pp=2e5(_H-y)/_Hglobal ana_dis5 = -2e5*_H/(2.0*al1) ; new pp - old pp=-2e5(_H-y)/_Hglobal pnt = gp_near(1,0,10)global num_dis = gp_zdisp(pnt)

end

gen zone brick size 1 1 10model mech e fluid fl_isoprop bu @m_bu sh @m_shini dens @m_d; --- (column is dry) ---ini sat 0; --- boundary conditions ---fix z range z 0fix x range x 0fix x range x 1fix y range y 0fix y range y 1; --- gravity ---set grav 0 0 @mgrav; --- histories ---his add fish @num_dishis add fish @ana_dis1

; --- initial equilibriumset fluid off mech onsolvelist @num_dis @ana_dis1save ini

; -------------------------------------------------; --- Fully coupled ---; -------------------------------------------------restore ini

FLAC3D Version 4.0


title ’Buoyancy - fully coupled’ini xdis 0 ydis 0 zdis 0his reset; --- histories ---his add fish @num_dishis add fish @ana_dis2his add fltime

; --- add water ---ini sat 1ini fden=@w_d fmod=2e8 ftens=-1e10prop poro=@m_n perm 1e-8; --- boundary conditions ---fix pp 0 range z 10; --- static equilibrium ---set fluid on mech onset mech force 0set mech ratio 1e-6set mech substep 10 autoset fluid substep 256; --- we can run this simulation coupled, using ---solve age 300list @num_dis @ana_dis2save ex1a

; -------------------------------------------------; --- Uncoupled ---; -------------------------------------------------restore inititle ’Buoyancy - uncoupled’ini xdis 0 ydis 0 zdis 0his reset; --- add water ---ini sat 1ini fden=@w_d fmod=2e8 ftens=-1e10prop poro=@m_n perm 1e-8; --- boundary conditions ---fix pp 0 range z 10; --- static equilibrium ---set fluid on mech offset mech force 0set mech ratio 1e-6; --- we can run this simulation uncoupled, using ---solve age 300

; --- histories ---

FLAC3D Version 4.0


his add fish @num_dishis add fish @ana_dis2

set flow off mech onini fmod 0solvelist @num_dis @ana_dis2save ex1b

; -------------------------------------------------; --- Sink it deeper, under 20m of water ---; -------------------------------------------------restore ex1bini fmod 2e8title ’Sink it deeper, under 20m of water’ini xdis 0 ydis 0 zdis 0his reset; --- fluid boundary conditions ---fix pp 2e5 range z 10; --- apply pressure of water ---apply nstress -2e5 range z 10; --- static equilibrium ---; --- we can run this simulation uncoupled, using ---set flow on mech offsolve age 600

; --- histories ---his add fish @num_dishis add fish @ana_dis3

set flow off mech onini fmod 0solvelist @num_dis @ana_dis3save ex1c

; -------------------------------------------------; --- Seepage force: flush up ---; -------------------------------------------------restore ex1cini fmod 2e8title ’Seepage force: flush up’ini xdis 0 ydis 0 zdis 0his reset; --- flush fluid up ---fix pp 5e5 range z 0

FLAC3D Version 4.0


; --- static equilibrium ---; --- we can run this simulation uncoupled, using ---set flow on mech offsolve age 1200

; --- histories ---his add fish @num_dishis add fish @ana_dis4

set flow off mech onini fmod 0

solvelist @num_dis @ana_dis4save ex1e

; -------------------------------------------------; --- Seepage force: flush down ---; -------------------------------------------------restore ex1cini fmod 2e8title ’Seepage force: flush down’ini xdis 0 ydis 0 zdis 0his reset; --- flush fluid up ---fix pp 1e5 range z 0; --- static equilibrium ---; --- we can run this simulation uncoupled, using ---set flow on mech offsolve age 1200; --- histories ---his add fish @num_dishis add fish @ana_dis5

set flow off mech onini fmod 0solvelist @num_dis @ana_dis5save ex1f

FLAC3D Version 4.0


1.9.2 Pore Pressure Initialization and Deformation

In FLAC3D, when equilibrium stresses are initialized in the model with the INITIAL command (follow-ing, for example, the procedure as described in Section 3.4.2 in the User’s Guide) and mechanicalsteps are taken, no stress increment is calculated by the code, and thus no displacement is gener-ated, because the model is in equilibrium with the stress boundary conditions and applied loads.In other words, FLAC3D does not calculate the deformations associated with installation of equi-librium stresses, when using the INITIAL command. If the stress state is installed in this way at thebeginning of a run, the initial stress state is taken as the reference state for displacements. (This isworth noting because the logic may be different from other codes, in which the zero stress state istaken as reference for calculation of displacements.)

The situation regarding pore pressure in a fluid-mechanical simulation is similar: by default, if porepressures are initialized with the INITIAL pp command and are in equilibrium with the fluid boundaryconditions and hydraulic loading, and fluid steps are taken, then, by default, no increment of porepressure will be generated by the code. If the INITIAL pp command is issued at the beginning of therun, this initial state is taken as the reference state for pore pressure. There will be no stress change(and, if mechanical steps are taken, no displacement) as a result of the pore pressure initialization,because no increment of pore pressure is calculated by the code. In other words, by default, FLAC3D

does not calculate the deformation associated with installation of equilibrium pore pressures whenusing the INITIAL pp command.

This only applies, by default, to equilibrium pore pressures established using the INITIAL pp com-mand, the WATER table command or a FISH function to initialize pore pressures. Pore pressurechange that is calculated by FLAC3D, on the other hand, will always generate stress change; if thesystem is brought out of equilibrium by the stress change and mechanical steps are taken, thendeformations will be generated, if conditions allow.

If the deformation associated with a new distribution of pore pressure, assigned using the INITIALpp command, the WATER table command or a FISH function to initialize pore pressures, needs tobe calculated, then a special stress-correction technique needs to be used. This technique consistsof: 1) subtracting from the total normal stresses the increment of pore pressure in the zones affectedby the change, multiplied by Biot coefficient (the pore pressure increment in a zone is calculatedby averaging nodal values); 2) adjusting the saturation to zero in the dry region, and to one in theregion filled with fluid; 3) adjusting the input material density to the bulk value, above and belowthe phreatic surface (in case it exists), and the simulation is run using CONFIG fluid; and 4) cyclingthe model to mechanical equilibrium. A simple example illustrating the heave of a soil layer usingthis technique is presented below.

For the example, we consider a layer of soil of large lateral extent, and thickness H = 10 meters,resting on a rigid base. The layer is elastic, the drained bulk modulus,K , is 100 MPa, and the shearmodulus,G, is 30 MPa. The bulk density of the dry soil, ρ, is 1800 kg/m3, and the density of water,ρw, is 1000 kg/m3. The porosity, n, is uniform; the value is 0.5. And gravity is set to 10 m/sec2.

Initially, the water table is at the bottom of the layer, and the layer is in equilibrium under gravity.We evaluate the heave of the layer when the water level is raised to the soil surface. This simpleproblem is similar to the one analyzed in Section 1.9.1.1. The difference is, here, we do not require

FLAC3D Version 4.0


the code to find the new pore pressure distribution. The simulation can be carried out with orwithout using the groundwater configuration (CONFIG fluid). We consider both cases.

The grid for this example contains 20 zones: 10 in the z-direction, 2 in the x-direction, and 1 in they-direction. The origin of axes is at the bottom of the model. The mechanical boundary conditionscorrespond 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 INITIAL sxx,INITIAL syy and INITIAL szz commands, using a value of 0.5714 (equal to (K− 2G/3)/(K+ 4G/3))for the coefficient of earth pressure at rest, ko.

There are two competing effects on deformation associated with raising the water level: first, theincrease of pore pressure will generate heave of the layer; and second, the increase in soil bulkdensity due to the presence of the water in the pores will induce settlement. To model the combinedeffects on deformation, of a rise in water level up to the soil surface, we proceed as follows.

If we do not use CONFIG fluid, we specify a hydrostatic pore-pressure distribution corresponding tothe new water level by using either the INITIAL pp command or the WATER table command, and wespecify a wet bulk density for the soil beneath the new water level. We apply the stress correctiondescribed under point 1) above, using INI sxx add, INI syy add and INI szz add. Finally, we cycle themodel to static equilibrium.

If we do use CONFIG fluid, we again specify a hydrostatic pore-pressure distribution correspondingto the new water level using the INITIAL pp or WATER table command. The saturation is initializedto 1 below the water level, and to zero above. However, we do not update the soil density to accountfor the presence of water beneath the new water level (this is done automatically by the code). Weagain perform a stress correction, as described earlier. Finally, we SET flow off and cycle the modelto static equilibrium.

The final response is identical for both cases. The plot of displacement vectors in Figure 1.59indicates that the rise of the water table has induced a heave of the soil layer. The surface heave,uh, can be evaluated analytically using Eq. (1.146),

uh = (1 − n)ρwg

2α1H 2 (1.146)

where α1 = K + 4G/3. The theoretical value for uh is 1.786 × 10−3 meters. The theoretical andnumerical values compare well. The data files for the simulations are listed in Examples 1.22 and1.23.

FLAC3D Version 4.0


Figure 1.59 Heave of a soil layer

The project file for this example, “PPInitAndDeformation.f3prj,” is located in folder “datafiles\fluid\PPInitAndDeformation.”

Example 1.22 Heave of a soil layer, without config fluid


title ’Raising the water table - not in config fluid’

define setupglobal m_bu = 1e8 ; drained bulk modulusglobal m_sh = 0.3e8 ; shear modulusglobal m_d = 1800. ; material dry mass densityglobal m_n = 0.5 ; porosityglobal w_d = 1000. ; water mass densityglobal _grav = 10. ; gravityglobal _H = 10. ; height of column

; --- derived quantities ---global m_rho = m_d+m_n*w_d ; material bulk wet densityglobal mgrav = -_grav

end@setupdefine num_dis

FLAC3D Version 4.0


local al1 = m_bu+4.0*m_sh/3.0global ana_dis = -(m_n-1.0)*w_d*_grav*_Hˆ2/(2.0*al1)global pnt = gp_near(1,0,10)global num_dis = gp_zdisp(pnt)

end

gen zone brick p0 (0,0,0) P1 (2,0,0) P2 (0,1,0) P3 (0,0,10) &size 2 1 10 ratio 1 1 1

model mech elasprop bu @m_bu sh @m_sh; --- column is dry ---prop density @m_d; --- boundary conditions ---fix z range z 0fix x range x 0fix x range x 2fix y range y 0fix y range y 1; --- gravity ---set grav 0 0 @mgrav; --- histories ---his nstep 10his add fish @num_dishis add fish @ana_dis; --- initial equilibrium ---ini szz -1.8e5 grad 0 0 1.8e4ini sxx -1.029e5 grad 0 0 1.029e4ini syy -1.029e5 grad 0 0 1.029e4;set mech ratio 1e-6solvesave ini

; --- raise water level ---; (note: when not in CONFIG FLUID, water density is assigned using:)water density @w_d; (we can do it this way ...)ini pp 1e5 grad 0 0 -1e4; (or this way ...);water table origin 0 0 _H normal 0 0 -1; (total stress adjustement)ini sxx add -1e5 grad 0 0 1e4ini syy add -1e5 grad 0 0 1e4ini szz add -1e5 grad 0 0 1e4; --- use wet density below water table ---prop dens @m_rho

FLAC3D Version 4.0


; --- static equilibrium ---solvelist @num_dis @ana_dissave ncflreturn

Example 1.23 Heave of a soil layer, with config fluid


config fluid

title ’Raising the water table - config fluid’define setup

global m_bu = 1e8 ; drained bulk modulusglobal m_sh = 0.3e8 ; shear modulusglobal m_d = 1800. ; material dry mass densityglobal m_n = 0.5 ; porosityglobal w_d = 1000. ; water mass densityglobal _grav = 10. ; gravityglobal _H = 10. ; height of column

; --- derived quantities ---global m_rho = m_d+m_n*w_d ; material bulk wet densityglobal mgrav = -_grav

end@setupdefine num_dis

local al1 = m_bu+4.0*m_sh/3.0global ana_dis = -(m_n-1.0)*w_d*_grav*_Hˆ2/(2.0*al1)global pnt = gp_near(1,0,10)global num_dis = gp_zdisp(pnt)

end

gen zone brick p0 (0,0,0) P1 (2,0,0) P2 (0,1,0) P3 (0,0,10) &size 2 1 10 ratio 1 1 1

model mech elas fluid fl_isoprop bu @m_bu sh @m_sh

; --- column is dry ---; (initialize sat at 0)ini sat 0prop density @m_d; --- boundary conditions ---fix z range z 0

FLAC3D Version 4.0


fix x range x 0fix x range x 2fix y range y 0fix y range y 1; --- gravity ---set grav 0 0 @mgrav; --- histories ---his nstep 10his add fish @num_dishis add fish @ana_dis; --- initial equilibrium ---ini szz -1.8e5 grad 0 0 1.8e4ini sxx -1.029e5 grad 0 0 1.029e4ini syy -1.029e5 grad 0 0 1.029e4;set mech ratio 1e-5set fluid off mech onini fmod 0solvesave ini

; --- raise water level ---; (initialize sat at 1 below the water level)ini sat 1; (note: in CONFIG FLUID, water density is assigned using:)ini fdens @w_d; (we can do it this way ...)ini pp 1e5 grad 0 0 -1e4; (or this way ...); (note: we can use water table command in config fluid,; this is different from FLAC);water table origin 0 0 _H normal 0 0 -1; (total stress adjustement)ini sxx add -1e5 grad 0 0 1e4ini syy add -1e5 grad 0 0 1e4ini szz add -1e5 grad 0 0 1e4; --- Note: no need to specify wet density below water table ---; --- static equilibrium ---set fluid off mech onini fmod 0solvelist @num_dis @ana_dissave cflreturn

FLAC3D Version 4.0


1.9.3 Effect of the Biot Coefficient

The Biot coefficient, α, relates the compressibility of the grains to that of the drained bulk material:

α = 1 − K

Ks(1.147)

where K is the drained bulk modulus of the matrix, and Ks is the bulk modulus of the grains (seeDetournay and Cheng 1993, for reference).

For soils, matrix compliance is usually much higher than grain compliance (i.e., 1/K >>> 1/Ks),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 ofmagnitude and, as a result, the Biot coefficient may be almost zero. Consider, for example, asample of porous elastic rock. The pores are saturated with fluid at a pressure, p, and a total externalpressure, P , is applied around the periphery (i.e., on the outside of an impermeable sleeve). Theproblem can be analyzed by superposition of two stress states: state a, in which fluid pressure andexternal pressure are both equal to p; and state b, in which pore pressure is zero, and the externalpressure is P − p (see Figure 1.60).

Figure 1.60 Decomposition of stresses acting on a porous, elastic rock

The stress-strain relation for state a may be expressed as

p = Ksεa (1.148)

For state b (there is no fluid), we can write

P − p = Kεb (1.149)

FLAC3D Version 4.0


The total strain is given by superposition of the strain in state a and in state b:

ε = εa + εb (1.150)

After substitution of εa from Eq. (1.148), and εb from Eq. (1.149), we obtain

ε = p

Ks+ P − p

K(1.151)

After some manipulations, the stress-strain equation takes the form

P −[

1 − K

Ks

]

p = P − αp = Kε (1.152)

Clearly, then, in the framework of Biot theory, a zero Biot coefficient implies that the elasticstress-strain law becomes independent of pore pressure. Of course, in general, porous rocks donot behave elastically, and pore pressure has an effect on failure. Also, if fluid flow in rocksoccurs mainly in fractures, Biot theory may not be applicable. Nonetheless, there are numerousinstances where the small value of the Biot coefficient may help explain why pore pressure haslittle effect on deformation for solid, porous (i.e., unfractured) rocks. (For example, the effect onsurface settlement of the raising or lowering of the water table in a solid porous rock mass may beunnoticeable.)

Note that the preceding discussion addresses only one of the effects of grain compressibility. TheBiot coefficient also enters the fluid constitutive law, which relates change of fluid content tovolumetric strain.

The logic for grain compressibility, as developed in the framework of Biot theory, is provided inFLAC3D. Simple verification examples are described below to illustrate the logic.

For reference, in the examples below, in the special case εyy = 0, the principal stress-strain relationshave the form

σxx + α p = (K + 4G/3) εxx + (K − 2G/3) εzz (1.153)

σzz + α p = (K + 4G/3) εzz + (K − 2G/3) εxx (1.154)

p = M( ξ − α εv) (1.155)

where ξ is the variation of fluid content per unit volume of porous media, and εv is the incre-mental volumetric strain.

FLAC3D Version 4.0


1.9.3.1 Undrained Oedometer Test

An undrained oedometer test is conducted on a saturated poro-elastic sample. The Biot modulusis 1 MPa, and the Biot coefficient is 0.3 for this test. The model is a single square zone of unitdimensions with roller boundaries on the sides and bottom. A constant velocity, v, is applied to thetop. Fluid flow is turned off, and the simulation is run for 10 calculational steps. The SET biot oncommand is given to select the Biot modulus and coefficient rather than the fluid bulk modulus.

Because the sample is laterally confined, εxx = εyy = 0 and εv = εzz. For undrainedconditions, ξ = 0. The analytical value for pore pressure (see Eq. (1.155)) is then

p = −αM εzz (1.156)

The analytical stresses are obtained by substituting pore pressure and strain components intoEqs. (1.153) and (1.154):

σxx =[

(K − 2G/3)+ α2M]

εzz (1.157)

σzz =[

(K + 4G/3)+ α2M]

εzz (1.158)

After 10 calculation steps, εzz = −10v. The agreement between analytical and numerical valuesfor pore pressure and stresses is checked with the FISH function checkit. The analytical andnumerical results are identical. Example 1.24 lists the data file for this example. The correspondingproject file, “oedometer.f3prj,” is located in folder “datafiles\fluid\oedometer.”

Example 1.24 Undrained oedometer test

; file for ’Undrained Oedometer test’new projectset fish autocreate off

config fluid

title ’Undrained Oedometer test’define setup

global c_b = 1e8 ; drained bulk modulusglobal c_s = 0.3e8 ; shear modulusglobal c_n = 0.5 ; porosityglobal c_a = 0.3 ; Biot coefficient (alpha)global c_bm = 1e8 ; Biot modulusglobal c_zv = -1e-3 ; z-velocityglobal c_ns = 10 ; number of steps

FLAC3D Version 4.0


end@setup

gen zone brick p0 (0,0,0) P1 (1,0,0) P2 (0,1,0) P3 (0,0,1) &size 1 1 1

model mech elas fluid fl_isoprop bu @c_b sh @c_sini ftens -1e10set fluid biot onprop poro=@c_n biot_c=@c_a; --- boundary conditions ---fix x y zini zvel=@c_zv range z 1; --- undrained compression ---ini biot_mod=@c_bmset fluid off mech onstep @c_nsreturn

1.9.3.2 Pore Pressure Generation in a Confined Sample

The effect of pore pressure generation is shown for the case of a confined sample in an imper-meable sleeve. The sample geometry and properties are the same as in the previous example, inSection 1.9.3.1. Roller boundaries are set on all four sides of the model. The boundaries are alsoimpermeable (by default). Fluid flow is turned on, and a volumetric water source with a unit flowrate is applied to the model to raise the pore pressure. The simulation is run for 10 fluid flow steps.

At the end of the simulation, ξ = 10 t . The grid is fully constrained, hence εxx = εyy = εzz = εv = 0. The analytical value for pore pressure is found, from Eq. (1.155), to be

p = M(10 t) (1.159)

The analytical stresses are then derived from Eqs. (1.153) and (1.154) to be

σxx = σyy = σzz = −αM(10 t) (1.160)

Numerical and analytical values for pore pressure and stresses are compared with the FISH functioncheckit, and the results are identical. Example 1.25 lists the data file. The corresponding projectfile, “ppgen-confined.f3prj,” is located in folder “datafiles\fluid\ppgen-confined.”

FLAC3D Version 4.0


Example 1.25 Pore pressure generation in a confined sample

; file for ’Pore pressure generation in a confined sample’


config fluid

title ’Pore pressure generation in a confined sample’

def setupglobal c_b = 1e8 ; drained bulk modulusglobal c_s = 0.3e8 ; shear modulusglobal c_n = 0.5 ; porosityglobal c_a = 0.3 ; Biot coefficient (alpha)global c_bm = 1e8 ; Biot modulusglobal c_k = 1e-10 ; mobilityglobal c_ws = 1.0 ; volumetric water sourceglobal c_ns = 10 ; number of steps

end@setupgen zone brick p0 (0,0,0) P1 (1,0,0) P2 (0,1,0) P3 (0,0,1) &

size 1 1 1model mech elas fluid fl_isoprop bu @c_b sh @c_sini ftens -1e10set fluid biot onprop poro=@c_n biot_c=@c_a perm=@c_kini biot_mod=@c_bm; --- boundary conditions ---fix x y z; --- water source ---apply vwell @c_wsset fluid on mech offstep @c_nsreturn

FLAC3D Version 4.0


1.9.3.3 Pore Pressure Generation in an Infinite Layer

This example is similar to the preceding confined sample, except that the top boundary is notconstrained. All boundaries are impermeable, and a volumetric water source with unit flow isapplied to raise the pore pressure. Both mechanical and fluid-flow calculations are turned on, andthe simulation is run for 10 fluid-flow steps. Note that 200 mechanical sub-steps are taken everyfluid step in order to keep the system in quasi-static equilibrium state.

For this example, εxx = εyy = 0, εv = εzz and σzz = 0. Using these conditions inEq. (1.154), we obtain

p = (K + 4G/3) εzzα

(1.161)

After substituting Eq. (1.161) for p in Eq. (1.155) and solving for εzz, we find

εzz = αM ξ

(K + 4G/3)+ α2M(1.162)

Analytical expressions for pore pressure and stress can now be derived from Eqs. (1.153) and(1.155):

p = M( ξ − α εzz) (1.163)

σxx = (K − 2G/3) εzz − α p (1.164)

Numerical and analytical values for vertical displacement, pore pressure and stresses are comparedwith the FISH function checkit, and the results are identical. Example 1.26 lists the data file.The corresponding project file, “ppgen-infinite.f3prj,” is located in folder “datafiles\fluid\ppgen-infinite.”

FLAC3D Version 4.0


Example 1.26 Pore pressure generation in an infinite layer

; file for ’Pore pressure generation in an infinite layer’


config fluid

title ’Pore pressure generation in an infinite layer’define setup

global c_b = 2.0 ; drained bulk modulusglobal c_s = 1.0 ; shear modulusglobal c_n = 0.5 ; porosityglobal c_a = 0.3 ; Biot coefficient (alpha)global c_bm = 1 ; Biot modulusglobal c_k = 1.0 ; mobilityglobal c_ws = 1. ; volumetric water sourceglobal c_ns = 10 ; number of steps

end@setupgen zone brick p0 (0,0,0) P1 (1,0,0) P2 (0,1,0) P3 (0,0,1) &

size 1 1 1model mech elas fluid fl_isoprop bu @c_b sh @c_sini ftens -1e10set fluid biot onprop poro=@c_n biot_c=@c_a perm=@c_kini biot_mod=@c_bm; --- boundary conditions ---fix x yfix z range z 0; --- water source ---apply vwell @c_wsset fluid on mech onset mech force 0 mech ratio 1e-10set fluid substep 1set mech substep 200 autosolve age 1.6return

FLAC3D Version 4.0


1.9.4 Semi-confined Aquifer

Fluid leakage into a shallow semi-confined aquifer can be modeled with FLAC3D using the APPLYleakage command. This is demonstrated for the example defined by the sketch in Figure 1.61. Theaquifer has a length L, height H , and rests on an impermeable base. Fluid flow obeys Darcy’slaw; the mobility coefficient k is homogeneous and isotropic. The semi-permeable top layer haspermeability k∗, and thicknessH∗. The effect of gravity is neglected in this example. Fluid pressureat the top of the leaky layer is constant and equal to p∗. The lateral fluid-flow conditions correspondto a constant pressure p0 at the left boundary, and p1 at the right.

The objective is to determine the steady-state pore-pressure profile and total leakage into the aquifer.The general solution of pore pressure for a shallow semi-confined aquifer has the form (see Strack1989)

p − p∗ = Aex/λ + Be−x/λ (1.165)

where λ is the seepage factor, which has the dimension of length and is defined as λ = √kHH∗/k∗;

A and B are constants determined from the pressure boundary conditions.

Figure 1.61 Shallow semi-confined aquifer

The boundary conditions for this problem are

p = p0 at x = 0

p = p1 at x = L

FLAC3D Version 4.0


The pore pressure solution is

p = Aex/λ + Be−x/λ + p∗ (1.166)

where

A = (p1 − p∗)eL/λ − (p0 − p∗)e2L/λ − 1

B = (p0 − p∗)e2L/λ − (p1 − p∗)eL/λ

e2L/λ − 1

The steady state discharge over the height of the aquifer is obtained from Darcy’s law:

Qx = kHdp

dx(1.167)

After differentiation of Eq. (1.166) with respect to x, and substitution into Eq. (1.167), we obtain

Q = kH

λ

[

Aex/λ − Be−x/λ]

(1.168)

The total amount of leakage into the aquifer is, by continuity of flow, equal to the difference betweenthe discharge leaving at x = L and that entering at x = 0. Using Eq. (1.168), we obtain, after somemanipulation,

Qx = 2kH

λ

eL/λ − 1

eL/λ + 1

[p0 + p1

2− p∗

]

(1.169)

Eqs. (1.166) and (1.169) are used for comparison to the FLAC3D solution.

The FLAC3D data file for this problem is listed in Example 1.26. The analytical solution is pro-grammed in FISH as part of the data file. The FLAC3D model is a 20 zone by 2 zone mesh witha constant pore pressure of p0 = 20 kPa applied at the left boundary, x = 0, and a constant porepressure of p1 = 10 kPa applied at the right boundary, x = 20 m. A leaky aquifer boundary conditionis applied along the top boundary of the model, y = 1 m, using the APPLY leakage command. Thepore pressure at the top is p∗ = 1.8 kPa, and the leakage coefficient h (see Eq. (1.18)), is evaluatedto be k∗/H∗ = 2.98 × 10−9 m3 /(N sec), based on the properties of the leaky layer. The propertiesfor this problem are listed in the ini h4 function in Example 1.26. The fluid-flow calculationmode is turned on, the mechanical calculation mode is turned off, and the simulation is run untilsteady-state flow is reached.

FLAC3D Version 4.0


The FISH function tab pp compares the amount of leakage calculated by FLAC3D to the solutionof Eq. (1.169) at steady-state flow. The difference is printed (in a FISH dialog message) to be0.03%. The analytical and numerical pore pressure profiles recorded along the base of the model,from x = 0 to x = 20, are compared in Figure 1.62:

Figure 1.62 Pore pressure profile

The project file for this example, “semi-confAquifer.f3prj,” is located in folder “datafiles\fluid\semi-confAquifer.”

Example 1.27 Shallow confined aquifer with leaky boundary

; file for ’Semiconfined aquifer’new projectset fish autocreate off

config fluid

title ’Semiconfined aquifer’

define setup; --- aquifer ---

global c_h = 1. ; heightglobal c_l = 20. ; lateral extent

FLAC3D Version 4.0


global c_n = 0.4 ; porosityglobal c_k = 2e-8 ; mobilityglobal c_wb = 2e9 ; water bulk modulusglobal c_p0 = 2e4 ; upstream ppglobal c_p1 = 1e4 ; downstream pp

; --- leaky layer ---global c_hs = 0.1 ; heightglobal c_ks = 3e-10 ; mobilityglobal c_ps = 1.8e4 ; pp

; --- utility ---global _eps = 0.1 ; small length compared to zone size

; --- derived value ---global c_cs = c_ks/c_hs ; leakage coefficientglobal zm = c_h-_epsglobal zp = c_h+_epsglobal xm = c_l-_eps

end@setupgen zone brick p0 (0,0,0) P1 (@c_l,0,0) P2 (0,1,0) P3 (0,0,@c_h) &

size 20 2 2model fluid fl_isoini ftens -1e10prop poro=@c_n perm=@c_kini fmod=@c_wb ftens -1e10; --- boundary conditions ---apply leakage @c_ps @c_cs range z @zm @zp ;c_hfix pp @c_p0 range x 0fix pp @c_p1 range x @c_l; --- fluid flow solution ---set fluid on mech offsolve age 1;save confaquiferreturn

FLAC3D Version 4.0


1.10 Input Instructions for Fluid-Flow Analysis

This section summarizes all of the FLAC3D commands and FISH variables related to fluid-flowanalysis. See Section 1 in the Command Reference and Section 1 in the Plot Command Referencefor a complete listing of FLAC3D commands, and Section 2.5 in the FISH volume for a listing ofFLAC3D -specific variables accessed by FISH.

1.10.1 FLAC3D Commands

The following commands are provided to run fluid-flow problems. Note that the fluid-flow com-mands are invoked by keywords used with existing commands for a standard mechanical analysis.For a transient fluid-flow analysis, the CONFIG fluid command must be the first fluid-flow commandgiven, before any other fluid-flow commands are invoked. CONFIG fluid does not have to be givenfirst in a data file, but can be specified at the point that the fluid-flow calculation is to begin.

APPLY keyword <keyword> value <keyword> <range . . . >

The APPLY command is used to apply fluid-flow boundary conditions to any externalor internal boundary of the model grid, or to interior gridpoints. The user must specifythe keyword type to be applied (e.g., discharge), the numerical value (if required),and the range over which the boundary conditions are to be applied. The range can begiven in several forms (see Section 1.1.3 in the Command Reference). If no rangeis specified, then the command applies to the entire model. Optional keywords mayprecede or follow the numerical value. The optional keywords are described for theAPPLY command in Section 1.3 in the Command Reference.

Three keyword types are used to apply fluid-flow boundary conditions. The associ-ated keywords are given for each type:

Gridpoint-Type Keywords

pp v <interior>

A fluid pore pressure, v, is applied at the boundary gridpoints. (Notethat this is not a mechanical boundary condition. Use the APPLYnstress command to specify a mechanical pressure boundary condi-tion.) Use the interior keyword to apply a pore pressure to an interiorgridpoint.

pwell v <interior>

A fluid-flow rate, v, (e.g., in m3/sec) is applied at each boundarygridpoint in the specified range. This command is used to specify aconstant inflow (v > 0) or outflow (v < 0) along a fluid-flow boundary.Use the interior keyword to apply the condition to an interior gridpoint.When a new well is applied to a gridpoint with an existing well, thenew well flow rate replaces the existing well flow-rate.

FLAC3D Version 4.0


Face-Type Keywords

discharge v

Fluid flux v is the component of the specific discharge vector (e.g., in[m/sec]) applied normal to the boundary.

leakage v1 v2

v1 is the pore pressure in the leaky layer.

v2 is the leakage coefficient, h (e.g., in m3/N sec).

See Eq. (1.18) for the formula for a leaky boundary condition. Aleaky condition is applied over the range of faces specified. Thehistory keyword is not active for leakage.

Zone-Type Keywords

vwell v

A volume rate of flow, v (i.e., fluid volume per zone volume per unittime), is specified for each zone in the specified range (v > 0 forinflow). When a new volumetric source is applied to a zone with anexisting source, the new source replaces the existing source.

CONFIG fluid

This command specifies extra memory to be assigned to each zoneor gridpoint for a fluid-flow analysis. CONFIG fluid can be given atany stage of a FLAC3D analysis, but it must be given before any otherfluid-flow commands are invoked.

FIX pp <value> <range . . . >

The pore pressure is fixed at points in the gridpoint range. If a valueis given, the pore pressure is fixed at that value.

FREE pp <range . . . >

The pore pressure at points in the gridpoint range is allowed to change.

FLAC3D Version 4.0


HISTORY add <id nh> <nstep = n> keyword . . . x y z

The variables available for sampling are identified by the following keywords:

gp pp

Pore-pressure variables are sampled at gridpoint x,y,z.

zone pp

Pore-pressure variables are sampled at zone x, y, z.

History of Real Time

fltime creates a history of real time for fluid-flow problems.

INITIAL keyword <keyword> value <grad gx gy gz> <range . . . >

The following keywords apply. Units for fluid-flow properties are listed in Table 2.7in the User’s Guide.

biot mod Biot modulus, M , is initialized to the given value at all gridpoints inthe range specified. The Biot modulus is applied for the fluid modeinvolving compressible grains (SET fluid biot on). This property onlyapplies for CONFIG fluid mode.

fdensity The fluid mass density, ρf , is initialized to the given value at all zonesin the range specified. This property only applies for CONFIG fluidmode. Use the WATER density command in non-CONFIG fluid mode.

fmodulus The fluid modulus,Kf , is initialized to the given value at all gridpointsin the range specified. The fluid modulus is applied for the fluid modeinvolving incompressible grains (SET fluid biot off). This propertyonly applies for CONFIG fluid mode.

ftens The fluid tension limit is initialized to the given (negative) value atall gridpoints in the range specified. This property only applies forCONFIG fluid mode. (The default value is −10−15 units.)

pp The pore pressure is initialized to the given value at all gridpoints inthe range specified.

saturation The fluid saturation is initialized to the given value at all gridpointsin the range specified. (By default, value = 1 for fully saturatedzones when CONFIG fluid is specified.) This property only applies forCONFIG fluid mode.

FLAC3D Version 4.0


LIST keyword <keyword> . . . <range . . . >

This command prints various fluid-flow variables. The following keywords apply:

gp keyword

gridpoint data. A keyword is specified to print selected gridpoint datapertaining to fluid flow. The following keywords apply:

biot modulus gridpoint Biot modulus

fmodulus gridpoint fluid modulus

ftens gridpoint fluid tension limit

pp gridpoint pore pressure

saturation gridpoint saturation

zone keyword

zone data. A keyword is specified to print selected zone data pertain-ing to the fluid flow. The following keywords apply:

fdensity zone fluid density

pp zone pore pressure (average from gridpoint values)

property keyword

fluid-flow properties assigned to zones. Values areprinted for the property keyword. The keywords avail-able are listed below for the PROPERTY command.

MODEL keyword <range . . . >

This command associates a fluid-flow model with a range of zones.

fluid keyword

fl anisotropic anisotropic fluid flow

fl isotropic isotropic fluid flow

fl null Zone is null for fluid flow. (Null zones model imper-meable material.) Note that zones made null mechan-ically are not automatically made null for fluid flow.

FLAC3D Version 4.0


PROPERTY keyword value <keyword . . . > <region i,j> <var vx vy> <i = i1,i2 j = j1,j2>

This command assigns properties for the fluid-flow model identified by the MODELcommand. The required keywords to specify properties are listed below. Units forfluid-flow properties are listed in Table 2.7 in the User’s Guide.

fl anisotropic

(1) biot c Biot coefficient, α(2) fdip principal permeability dip angle, dip

(3) fdd principal permeability dip direction angle, dd

(4) frot principal permeability rotation angle, rot

(5) k1 principal permeability value, k1(6) k2 principal permeability value, k2(7) k3 principal permeability value, k3(8) kxx xx-component of permeability tensor(9) kyy yy-component of permeability tensor(10) kzz zz-component of permeability tensor(11) kxy xy-component of permeability tensor(12) kxz xz-component of permeability tensor(13) kyx yx-component of permeability tensor(14) porosity porosity, n(15) u thc undrained thermal coefficient, β

fl isotropic

(1) biot c Biot coefficient, α(2) permeability isotropic permeability coefficient, k(3) porosity porosity, n(4) u thc undrained thermal coefficient, β

Note: The principal directions of permeability, corresponding to k1, k2, k3, form aright-handed system. The angles dip and dd are the dip angle and dip-direction angleof the plane in which k1 and k2 are defined. The dip angle is measured from theglobal xy-plane, positive down (in negative global z-direction). The dip-directionangle is the angle between the positive y-axis and the projection of the dip-directionvector on the xy-plane (positive clockwise from the global y-axis). The angle rot isthe rotation angle between the k1-axis and the dip-direction vector, in the k1-k2-plane(positive clockwise from dip-direction vector). See Figure 1.1.

FLAC3D Version 4.0


SET keyword <keyword value> . . .

This command is used to set parameters in a FLAC3D model. The following keywordsapply:

fluid keyword <keyword value> . . .

The following keywords apply:

age t

t is the fluid-flow time limit for the fluid-flow calcula-tion using the SOLVE command.

biot onoff

The fluid-flow calculation uses the Biot coefficient, α,and the Biot modulus, M , if on. If off, then the fluidmodulus,Kf , and porosity, n, are used, and α = 1. Thedefault is off.

dt t

t defines the fluid-flow timestep. This timestep must bespecified for the implicit-solution scheme. By default,FLAC3D calculates fluid-flow timestep automaticallyfor the explicit-solution scheme. This keyword allowsthe user to choose a different timestep. If FLAC3D

determines that the user-selected value is too large fornumerical stability, the timestep will be reduced to asuitable value when fluid-flow steps are taken. Thecalculation will not revert to the user-selected valueuntil another SET fluid dt command is issued.

implicit onoff

The implicit-solution scheme in the fluid-flow modelis turned on or off. The default is off.

FLAC3D Version 4.0



fluid keyword <keyword value> . . .

on

off

The fluid-flow calculation process is turned on or off.The fluid-flow process is on by default when the CON-FIG fluid command is given. The fluid-flow calcula-tion is turned off for a mechanical-only calculation ora thermal-only calculation.

ratio value

The fluid-flow ratio limit is set to value for the SOLVEcommand. The value of ratio is defined as(|outf low| − |inf low|) / (|outf low| + |inf low|)where inflow (outflow) is the total volume of fluid en-tering (leaving) the flow domain per unit time (appliedcontributions included). (Different forms of ratio canbe specified – see the SET ratio command.) When theratio falls below value during the calculation process,the fluid-flow calculation will stop. (By default, theratio limit is set to 1.0 ×10−5.)

step value

The maximum number of steps to be taken when theSOLVE command is issued is set to value. (By default,unlimited stepping is allowed.)

substep value <auto>

The number of fluid-flow sub-steps in a coupled fluidflow-mechanical calculation or a coupled thermal-mechanical-fluid flow calculation is set to value. (Thedefault is value = 1.) The fluid-flow calculation is iden-tified as the slave component in the fluid-flow mechan-ical process or in the thermal-mechanical-fluid flowprocess when the optional keyword auto is given.

FLAC3D Version 4.0



time t

Fluid time is initialized to t.

mechanical keyword <keyword value> . . .

This command sets parameters for the mechanical calculationin a coupled mechanical-fluid-flow analysis. The followingkeywords apply:

force value

The out-of-balance force limit is set to value for theSOLVE command. When the maximum out-of-balanceforce falls below this limit, the mechanical calculationwill stop. (By default, the out-of-balance force limitis zero.)

on

off

The mechanical calculation process is turned on oroff. The mechanical process is on by default. Themechanical calculation is turned off for a flow-onlycalculation.

ratio value

The force ratio limit is set to value for the SOLVE com-mand. By default, ratio is defined as the average unbal-anced force magnitude for all gridpoints in the model,divided by the average applied force magnitude for allthe gridpoints. (Different forms of ratio can be spec-ified – see the SET ratio command.) When the ratiofalls below value during the calculation process, themechanical calculation will stop. (By default, the ratiolimit is set to 1.0 ×10−5.)

step value

The maximum number of steps to be taken when theSOLVE command is issued is set to value. (By default,unlimited stepping is allowed.)

FLAC3D Version 4.0



mechanical keyword <keyword value> . . .

substep value <auto>

The number of mechanical sub-steps in a coupledfluid-flow mechanical calculation or a coupledthermal-mechanical fluid-flow calculation is set tovalue. (The default is value = 100.) The mechani-cal calculation is identified as the slave component inthe fluid-flow mechanical process when the optionalkeyword auto is given.

SOLVE keyword value <keyword value> . . .

This command controls the automatic timestepping for fluid-flow and coupled fluid-flow mechanical and thermal fluid-flow mechanical calculations. A calculation isperformed until the limiting conditions, as defined by the following keywords, arereached.

age t

t is the maximum time limit for all processes involved in thecalculation.

clock t

t is the computer runtime limit, in minutes. (By default, thereis no limit on computer runtime.)

ratio value

value is the ratio limit for the active calculation process. (Bydefault, the limit is 1.0 × 10−5.)

WATER keyword value <keyword value> . . .

This command assigns a fixed groundwater table position and properties from which aconstant hydrostatic pore-pressure distribution is derived for use in an effective stresscalculation (not in CONFIG fluid mode). During calculation, FLAC3D uses effectivestresses (i.e., total stresses plus pore pressure) in constitutive models. Pore pressuresare defined at gridpoints. Zone pore pressures are calculated as the average of thezone gridpoints and are not stored. Pore pressures are not affected by zone volumechanges; nor is there any flow of water. Total stresses are displayed on plots andprintouts. Note that, when using this command, the saturated material density mustbe specified for zones below the water table, and the dry density for zones above.

FLAC3D Version 4.0


The following keywords apply:

density value

groundwater density, ρw [SI units; kg/m3]

table keyword value . . .

The WATER table command sets pore pressure for all gridpointsbelow the water table. The pore-pressure gradient is given bythe direction of the gravity vector, which can be arbitrary (seethe SET gravity command).

The water table plane can be defined in two forms: a singleinfinite plane, or an assembly of planar convex polygons. Foran infinite plane, the following keywords are used:

normal nx ny nz

normal direction to the plane, defined by unit vectornx, ny, nz and pointing in the direction of increasingpore pressure

origin x y z

one point at coordinate location (x y z) on the plane

Alternatively, the water table can be defined by an assemblyof planar, convex polygons. The following keyword phraseapplies:

face x1,y1,z1 . . . xn,yn,zn <face . . . >

The face polygon is defined by nodes x1, y1, z1 toxn, yn, zn. The nodes must produce a convex poly-gon. Faces can have any number of nodes but are splitinto triangles for storage. Only gridpoints that projectalong the gravity direction inside faces are assignedpore pressure. No checking of face overlapping orintersection is performed.

FLAC3D Version 4.0


1.10.2 FISH Variables

The following scalar variables are available in a FISH function to assist with fluid-flow analysis:

fldt fluid-flow timestep

fltime fluid-flow time

fluid ratio current fluid-flow ratio (set by the SET ratio command)

The following FLAC3D grid variables can be accessed and modified by a FISH function:

gp ftens gridpoint fluid tension limit

gp pp gridpoint pore pressure

gp sat gridpoint saturation

The following FLAC3D grid variable can be accessed but cannot be modified by a FISH function:

gp flow out-of-balance flow discharge at a gridpoint

The following FLAC3D zone variables can be accessed but cannot be modified by a FISH function:

z pp zone pore-pressure

z qx x-component of the specific discharge vector

z qy y-component of the specific discharge vector

z qz z-component of the specific discharge vector

Also, fluid-flow property values may be accessed (changed, as well as tested) in a FISH function.See the PROPERTY command in Section 1.10.1 for a list of the fluid-flow properties.

FLAC3D Version 4.0


1.11 References

Barenblatt, G. Dimensional Analysis. Gordon and Breach Science Publishers, 1987.

Bear, J. Dynamics of Fluids in Porous Media. New York: Dover, 1972.

Berchenko, I. Thermal Loading of a Saturated Rock Mass: Field Experiment and ModelingUsing Thermoporoelastic Singular Solutions. Ph.D. Thesis, University of Minnesota, 1998.

Biot, M. A. “General Solutions of the Equations of Elasticity and Consolidation for a PorousMaterial,” J. Appl. Mech., Trans. ASME, 78, 91-96, 1956.

Carslaw H. S., and J. C. Jaeger. Conduction of Heat in Solids, Second Edition. Oxford at theCalderon Press, 1959.

Chaney, R. C. “Saturation Effects on the Cyclic Strength of Sands,” in Proceedings: ASCEGeotechnical Engineering Division Specialty Conference (Pasadena, California, June 19-21,1978), Vol. 1, pp. 342-358, 1978.

Crank, J. The Mathematics of Diffusion, 2nd Ed. Oxford: Oxford University Press, 1975.

Dahlquist, G., and A. Bjorck. Numerical Methods. Prentice Hall, 1974.

Detournay, E., and A. H. D. Cheng. Comprehensive Rock Engineering. Pergamon Press Ltd.,1993.

Detournay, E., and A. H. D. Cheng. “Fundamentals of Poroelasticity,” in Comprehensive RockEngineering, Vol. 2, pp. 113-171. J. Hudson et al., eds. London: Pergamon Press, 1993.

Detournay, E., and A. H. D. Cheng. “Poroelastic Response of a Borehole in a Non-HydrostaticStress Field,” Int. J. Rock Mech. Sci. & Geomech. Abstr., 25(3), 171-182, 1988.

Harr, M. E. Groundwater and Seepage. Dover, 1991.

Karlekar, B. V., and R. M. Desmond. Heat Transfer, 2nd Ed. St. Paul: West Publishing Co., 1982.

Kochina, I., N. Mikhailov, and M. Filinov. “Groundwater Mound Damping,” Int. J. Engng. Sci.,21, 413-421, 1983.

Polubarinova-Kochina, P. Y. Theory of Groundwater Movement. Princeton: Princeton UniversityPress, 1962.

Stehfest, H. “Numerical Inversion of Laplace Transforms,” Communic. Ass. Comput. Mach., 13,47-49, 1970.

Taylor, D. W. Fundamentals of Soil Mechanics. New York: John Wiley, 1948.

Terzaghi, K. Theoretical Soil Mechanics. New York: John Wiley, 1943.

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Theis, C. V. “The Relation between the Lowering of the Piezometric Surface and the Rate andDuration of Discharge of a Well Using Groundwater Storage,” Trans. Am. Geophys. Union, 10,519-524, 1935.

Voller, V., S. Peng and Y. Chen. “Numerical Solution of Transient, Free Surface Problems in PorousMedia,” Int. J. Numer. Meth. Engng., pp. 2889-2906, 1996.

Wood, D. M. Soil Behaviour and Critical State Soil Mechanics. Cambridge: Cambridge Univer-sity Press, 1990.

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fluid mechanical interaction

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