Small and Large-strain1D/2D/3D Consolidation

Murray Fredlund, PhD, PEngSoilVision Systems Ltd.

Nov. 4rth, 2009Tailings and Mine Waste Conference

Banff, Canada

Overview• Introduction• Benchmarking / Verification• Why 2D and 3D analysis?• Layered tailings pit analysis• Conclusions

History• SOIL MECHANICS AND FOUNDATION ENGINEERING

EDUCATION IN 1949– Scope of field limited mainly to:

• Soil Classification• Capillarity and seepage• Stress analysis by elasticity• Consolidation and settlement analysis• Shear strength• Slope stability• Lateral pressures• Bearing capacity• Shallow and deep foundations

– Emphasis largely on saturated clays and sands

History• Tower of Pisa

• Consolidation problemshave been with us for awhile

Terzaghi Consolidation• Terzaghi proposed 1D small-strain

formulation a long time ago (1923, 1936)

• Problem is central to geotechnicalengineering practice

• Why has progress been so slow?– Coupling mechanism is inherently

mathematically very unstable

Core Problem• Need to solve

– Stress / deformation (Large-strain)– Fluid flow (continuity)

3. PWP increase

1. Apply load

2. Deformations4. PWP dissipates

5. Load transferred toeffective stress

Coupling• If true coupling is not properly handled

between the fluid andstress/deformation equations then theresults vary

• Uncoupled solutions do not produce thesame result as coupled solutions

• The difference between coupled anduncoupled in consolidation analysis issignificant

Terzaghi Consolidation• Formulation research has been slow at best

and is mathematically complex

• SoilVision research has been in the area of1D, 2D, 3D small and large strain

Researcher Type Saturation 1D 2D 3D

Terzaghi Small-strain Saturated

Biot; Mendel Small-strain Saturated

Fredlund & Dakshanamurthy Small-strain Unsaturated

Gibson; Schiffman; Townsend Large-strain

Formulations/Benchmarks• Significant work on small-strain coupled formulations

has been previously published by Biot (saturated),Mendel (saturated), Fredlund (unsaturated) andmany others

• Work on large-strain coupled consolidation has beenpublished by Schiffman, Gibson, Townsend, andothers

• Townsend published a series of 4 1D benchmarksand compared about 8 academic codes for eachbenchmark

• The Townsend benchmarks have been examined bySVS for the purposes of code comparison

Comparisons Uncoupled solutions will not include lateral

effects of deformation Mendel-Cryer effect

Comparisons Mandel-Cryer effect can be duplicated Varies based on Poisson’s Ratio

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.2 0.4 0.6 0.8 1.0

No

rm

al i

zed

Po

re

-Wa

t er

Pr

es

sur

e

Normalized Time

PR=0.49 PR=0.35 PR=0.05

Formulations• Formulations needed:

– 1D, 2D, 2D Axisymmetric, 3D• Elastic, Nonlinear elastic• Ksat, k as a function• Formulations are completed and working well

Elastic Non-linear elastic

Fixedmesh

Movingmesh

Fixedmesh HM

Movingmesh HM

Fixedmesh

Movingmesh

Fixedmesh HM

Movingmesh HM

1D

2D

2D Axisymmetric

3D

* HM - hydrological and mechanical coupling

Formulations/Benchmarks• Moving mesh / Lagrangian analysis• Difficult to find literature• Complex to benchmark results

No mesh updatingDeformation=0.5

Formulations - Uncoupled• Lagrangian

– Pure Lagrangiandeformation = 0.40

– Lagrangian-Eulariandeformation = 0.33

– Non-lagrangiandeformation =0.5

– Non-lagrangian OVERESTIMATESdeformations

Formulations/Benchmarks• Townsend scenario A: Time=1 year• Benchmark is reasonable

Formulations/Benchmarks• Townsend Scenario A: Time=1 year

Formulations/Benchmarks• Townsend Scenario A• Poisson’s ratio (0.3)• Ambiguity in boundary conditions

Solution - Runtimes• Expected run-times for numerical models are

important• The risk is that run-times will become too long

to complete projects in a reasonable time• Extended support for multi-processors has

been added• This has implications on speed

Non-linear compression example (uncoupled - moving mesh)v6.01

Dimension Nodes

Runtime(minutes) P4-Quad 2.4GHz -1 Core(s)

Runtime (minutes)P4-Quad 2.4GHz - 2Core(s)

Runtime (minutes)P4-Quad 2.4GHz - 4Core(s)

Time(minutes)/node

1D 201 0.62 1.09 1.22 0.00312D 1335 4.72 4.43 3.72 0.00282D Axisymmetric 1758 8.45 8.52 8.45 0.00483D 274 8.87 7.47 4.62 0.0168

Example Application – Pit• Sequenced tailings may be placed in

the pit as successive layers

Pit Filling• An example model in 2D

Differences

• Non-lagrangian solutions will OVER-ESTIMATE DEFORMATIONS– Easily demonstrated by SVS research

• Non-coupled solutions will most likelyUNDER-ESTIMATE PORE-WATERPRESSURES– If coupling is not properly performed it can

also lead to errors in solutions

Benefits• Benefits to the approach include:

– Formulation is theoretically correct anddefensible for reviewers

– Truly coupled solution can demonstrate theMendel-Cryer effect

– 2D and 3D solutions are stable anddemonstrate reasonable run-times

– Layered solutions work well– Reasonable for application to tailings projects

Thank you…!