AUBERTIN_Modeling arching effects in narrow backfilled stopes with FLAC.pdf

8/11/2019 AUBERTIN_Modeling arching effects in narrow backfilled stopes with FLAC.pdf

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1 INTRODUCTION

Even if backfill has been placed in undergroundstoping areas for many decades, it can be said that

backfilling still remains a growing trend in mining

operations around the world. This is particularly thecase in Canada where significant efforts have been

devoted, over the last twenty five years or so, to

improve our understanding of mining with backfill

(e.g., Nantel 1983, Udd 1989, Hassani & Archibald1998, Ouellet & Servant 2000, Belem et al. 2000,

2002).In recent years, the increased use of backfill in

mining has been fueled by environmental

considerations (e.g., Aubertin et al. 2002). Many

regulations now favor (and sometimes require) thatthe maximum quantity of wastes from the mine and

the mill be returned to underground workings. This

practice may induce significant advantages, as it canreduce the environmental impact of surface disposal

and the costs of waste management during mineoperation and upon closure.

The first purpose of mine backfill is nevertheless

to improve ground control conditions around stopes.

Various types of fills can be used to reach this goal,

each with its own characteristics. Backfill is oftenrequired to offer some self support properties, so itgenerally includes a significant proportion of binder

such as Portland cement and slag. But even the

strongest backfill is "soft" when compared to the

mechanical properties of the adjacent rock mass.

This difference in stiffness and yielding

characteristics between the two materials can be the

source of a stress redistribution in the backfill and

surrounding walls, as deformation of the backfillunder its own weight may create shear stresses along

the interface. For relatively narrow stopes, the load

transfer to the stiff abutments induce arching effects.

When this phenomenon occurs, the vertical stressbelow the main arching area tends to become

smaller than the backfill overburden pressure, as

shown by in situ measurements (e.g., Knutsson1981, Hustrulid et al. 1989).

The same type of arching behavior is also known

to occur in other types of structural systems, where arelatively soft material (like soil and grain) is placed

between stiff walls; examples include silos and bins

(Richards 1966, Cowin 1977, Blight 1986), ditches(Spangler & Handy 1984), and retaining walls (Hunt

1986, Take & Valsangkar 2001).

Arching effects and load redistribution can be

investigated using physical models, in situmeasurements, analytical solutions, and numerical

methods. The latter two approaches are particularly

attractive to identify the main influence factors, andto evaluate how these may affect the load

distribution in and around backfilled stopes.

In a companion paper, the authors have proposedsimple equations based on the Marston (1930)

theory to evaluate the load distribution in stope

backfill (Aubertin et al. 2003). The results of

analytical calculations have been compared tonumerical modeling performed with a commercially

available finite element code. The calculation results

Modeling arching effects in narrow backfilled stopes with FLAC

L.Li, M.Aubertin & R.Simoncole Polytechnique de Montral, Qc, Canada

B.Bussire & T.BelemUniversit du Qubec en Abitibi-Tmiscamingue, Qc, Canada

ABSTRACT: Numerical tools can be very useful to investigate the mechanical response of backfilled stopes.In this paper, the authors show preliminary results from calculations made with FLAC-2D. Its use isillustrated by showing the influence of stope geometry. The results are compared with analytical solutionsdeveloped to evaluate arching effects in backfill placed in narrow stopes. Some common trends are obtainedwith the two approaches, but there are also some differences regarding to magnitude of the stressredistribution induced by fill yielding.


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highlighted some important differences between the

two approaches, for the specific set of assumptions

adopted.In this paper, the authors use FLAC-2D (Itasca

2002) to further advance our understanding of the

load transfer process in and around narrow

backfilled stopes. In these calculations, some of theassumptions and input conditions differ from the

previous FEM calculations, including the use of a

somewhat more representative constitutive model

for the backfill. The mining sequence is also takeninto account. It is shown that for specific cases

amenable to analytical solutions, the calculatedresults from both approaches are fairly close to each

other.

2 ARCHING EFFECTS

Arching conditions can occur in geomaterials such

as soil, jointed rock mass and backfill, when

differential straining mobilizes shear strength whiletransferring part of the overburden stress to stiffer

structural components.

Arching typically occurs when portions of africtional material yields while the neighboring

material stays in place. As the yielding material

moves between stable zones, the relative movementwithin the former is opposed by shear resistance

along the interface with the latter. The shear stress

generated along the contact area tends to retain the

yielding material in its original position. This isaccompanied by a pressure reduction within the

yielding mass and by increased pressure on theadjacent stiffer material. Above the position of themain arch, a large fraction of the total overburden

weight in the yielding mass is transferred byfrictional forces to the unyielding ground on both

sides.

Investigations on models and in situmeasurements have shown that the magnitude of the

stress redistribution depends to a large extent on the

deformation of the walls confining the soft material(e.g., Bjerrum 1972, Hunt 1986).

A few analytical solutions have been developed toanalyze the pressure distribution since the

pioneering work of Janssen (1895) (see Terzaghi1943 for early geotechnical applications). Among

these, the Marston (1930) theory was proposed to

calculate the loads on conduits in ditches (see alsoMcCarthy 1988). The authors have used this theory

to develop an analytical solution for arching effects

in narrow backfilled stopes (Aubertin 1999).Figure 1 shows the loading components for a

vertical stope. On this figure, H is the backfillheight,Bthe stope width, and dhthe size of the layer

element; Wrepresents the backfill weight in the unitthickness layer. At position h, the horizontal layerelement is subjected to a lateral compressive forceC, a shearing force S, and the vertical forces VandV+dV.

H

B

V

V+ dV

SS

CC W

dh

h

dh

B

Backfill

stope

void space

rock mass

roc mass

layer element

Figure 1. Acting forces on an isolated layer in a vertical stope.

The force equilibrium equations for the layerelement provides an estimate of the stresses acting

across the stope (Aubertin et al. 2003). From these,

the vertical stress can be written as follow:

=

'tan2

)'tan/2exp(1v

BKh

K

Bh (1)

with

K= hh/vh (2)

where vh and hh are the vertical and horizontalstresses at depth h, respectively; represents the unitweight of the backfill; ' is the effective frictionangle between the wall and backfill, which is often

taken as the friction angle of the backfill, 'bf.Equations 1 and 2 constitute the Marston theory

solution. In this representation, K is the reactioncoefficient corresponding to the ratio of the

horizontal stress hh to vertical stress vh. Thisreaction coefficient depends on the horizontal wallmovement and material properties. When there is no

relative displacement of the walls, the fill is said to

be at rest, and the reaction coefficient is usuallygiven by (Jaky 1948):

K=K0= 1 sin'bf (3)

where 'bf is the friction angle of the backfill. Fortypical fill properties ('bf 30 to 35),K0is muchsmaller than unity.

If the walls move outward from the opening, the

horizontal pressure might be relieved, and the

reaction coefficient tends toward the active pressurecoefficient which can be expressed as (Bowles

1988):


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K=Ka= tan2(45 - 'bf/2) (4)

with Ka < K0. If an inward movement of the wallscompress the fill, it increases the internal pressure.Then, the reaction coefficient tends toward the

passive condition, which becomes (Bowles 1988):

K=Kp= tan2(45 + 'bf/2) (5)

withKp> 1 >K0.

In the above equations, it is assumed that cohesionis low in the backfill, so it can be neglected; the fill

then behaves as a granular material. A cohesion

would increaseKpbut decreaseKa.Figure 2 shows values of vh and hh calculated

with Equations 1 and 2 (with K = K0 = 0.5), andcalculated with the overburden pressure (i.e., vh=h and hh = K0vh). It can be seen that theoverburden stress represents the upper-bound

condition, which is applicable for low fill thickness

(or for wide stopes). Typically, when H 2 to 3B,

the pressure near the bottom of the stope becomesmore or less independent of the depth of the fill, in

accordance with measurements in bins (Cowin

1977).

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4h /B

stress(MPa)

forB = 6m

vh

hh

hh

vhoverburden

Marston theory

Figure 2. Overburden pressures are compared to vertical (vh)and horizontal (hh) stresses calculated with the Marston theory(Eqs. 1 - 2), with 'bf= 30, = 0.02 MN/m

3, andK=K0= 0.5.

3 NUMERICAL CALCULATIONS

3.1 Vertical stope

In a companion paper, the authors have shown

some calculation results obtained with a finite

element code (Aubertin et al. 2003). Significantdifferences have been revealed between the Marston

theory and these numerical calculations, which may

be explained, in part, by different assumptionsassociated to the two approaches. In this paper, the

same geometry and material properties (Fig. 3a) are

used for the basic calculations made with FLAC-2D.

The dimensions of the opening areH= 45m andB=6m, with a void of 0.5m left at the top of the stope.

The natural in situ vertical stress vin the rock massis obtained by considering the overburden weight

(for an overall depth of 250 m). The natural in situ

horizontal stress h is taken as twice the verticalstress v, which is a typical situation encountered inthe Canadian Shield. The rock mass is

homogeneous, isotropic and linear elastic, while the

granular backfill obeys a Coulomb criterion. Figure

3b shows the stress-strain relation used with theCoulomb plasticity model available in FLAC. This

constitutive behavior is different from the one usedfor the finite element calculations presented in

Aubertin et al. (2003), which was of the elastic-

brittle type. There are no interface elements in thecalculations made with FLAC (see discussion).

backfill

E= 300 MPa

v = 0.2

= 1800kg/m3

' = 30

c= 0 kPa

rock mass

(linear elastic)

E= 30 GPa

v= 0.3

= 2700kg/m3

vh= 2v

H=45m

B= 6 m

depth = 250 m

0.5m

x

y

void spacenatural stresses

rock massa)

1

1

b)

Figure 3. a) Narrow stope with backfill (not to scale) used formodeled with FLAC; the main properties for the rock mass and

backfill are given using classical geomechanical notations; b)

Schematic stress-strain behavior of the backfill (available as amaterial model in FLAC).

The mining and filling sequence is considered as

follow in the numerical modeling. The whole stope

is first excavated, and calculations are then

performed with FLAC-2D to an equilibrium state.


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Backfill is placed in the mined stope afterward, with

the initial displacement field set to zero when the

calculation is performed. In this manner, wallconvergence before backfilling is not included in the

calculations (this assumption is discussed in Section

4).

Figure 4 shows the vertical stress (Fig. 4a) andhorizontal stress (Fig. 4b) distribution in the

backfilled stope. As can be seen, the vertical and

horizontal stresses show a non uniform distribution.

At a given elevation, both stresses tend to be loweralong the wall than at the center. The stresses along

the central line increase more slowly than theoverburden pressures with depth. This indicates that

arching does take place in this backfilled stope.

a)

b)

Figure 4. Stress distribution in the backfilled stope calculated

with FLAC-2D: a) vertical stress yy; b) horizontal stress xx.

Figures 5 and 6 present modeling results for

stresses along the full height, with the overburden

and the Marston theory solutions. As expected, the

overburden stress is fairly close to analytical and

numerical results when the backfill depth is small.At larger depth, arching effects become important

and the vertical and horizontal stresses tend to be

lower than those due to the overburden weight of the

fill. However, the numerical results indicate that theMarston theory typically overestimates the amount

of stress transfer, hence underestimating the

magnitude of the vertical stress yy and of thehorizontal stress xxalong the stope central verticalline (Fig. 5). Along the walls (Fig. 6), the horizontal

stress is also underestimated by the Marston theory,

while the vertical stress component yy would beoverestimated for the active and at rest cases, withK= 1/2 or 1/3, respectively (and underestimated withK= 3, but the passive case is not representative ofthis system behavior).

0

0 .2

0 .4

0 .6

0 .8

0 9 18 27 36 45h (m)

yy

(MPa

mode ling with FLAC -2 D

overburden stressM arston theory

K = 1/3K = 1 /2

K = 3

a)

0

0.1

0.2

0.3

0 9 18 27 36 45

h (m)

xx

(MPa)

modeling with FLAC -2 D

overburden stress

Marston theory

K = 1/3

K = 1/2

K = 3

b)

Figure 5. Comparison of the stresses calculated along thevertical central line, at different elevations h, with theanalytical and numerical solutions: a) vertical stress yy; b)horizontal stress xx.

Figure 7 shows the stress distribution on the floor

of the stope, as obtained from the numerical andanalytical solutions. It can be seen that the

overburden pressure exceeds the stress magnitudes

given by the Marston theory (withK= 1/2 and 1/3),

which is in fair agreement with the numericalsimulations.


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0

0.2

0.4

0 .6

0 .8

0 9 18 27 36 45h (m)

yy

(MPa

modeling with FLAC-2D

overburden stress

M arston theory

K = 1/3K = 1/2

K = 3

a)

0

0.1

0.2

0 9 18 27 36 45h (m)

xx

(MPa)


overburden stress

M arston theory

K = 3

K = 1/3

K = 1/2

b)

Figure 6. Comparison of the stresses on the wall calculated at

different elevations h, with the analytical and numericalsolutions: a) vertical stress yy; b) horizontal stress xx.

3.2 Inclined stope

Mining stopes are rarely vertical. The inclinationof the foot-wall and hanging-wall may have a non-

negligible effect on the load distribution.

Figure 8 shows the geometry of an inclinedbackfilled stope modeled with FLAC-2D (a similar

stope was also modeled with the FEM code see

Aubertin et al. 2003). The rock mass and fill

properties as well as the in situ natural stresses areidentical to the previous case (see Fig. 3).

Figure 9 shows numerical calculations and results

based on overburden pressure and on the Marstontheory solution (without modification for

inclination). The horizontal stress calculated with

FLAC-2D along the inclined central line of the stopeis fairly close to the analytical solution (Fig. 9a), but

the vertical stress is underestimated by the Marston

theory (see Fig. 9b). Hence, modifications could berequired to apply such analytical approach to the

case of inclined stopes.

4 DISCUSSION

4.1 Influence of mining sequence

In the numerical calculations shown in a

companion paper (Aubertin et al. 2003), the mining

sequence was not taken into account, so the wallconvergence due to elastic straining of the rock mass

was imposed on the fill. This created an increase of

the mean stress in the fill, while vertical andhorizontal stresses locally exceeded the overburden

pressure and the Marston theory solution (near mid-

height of the stope).Modeling in this manner implies that the backfillis placed in the stope before wall displacement takes

place. For a single excavation stope, this is not a

realistic representation (at least for hard rockmasses). However, with a cut-and-fill mining

method where the mining slices (or layers) are

relatively small compared to the whole height of thestope, filling is usually made quickly after each cut.

In this case, wall convergence after each cutcompress the fill already in place (Knutsson 1981,

Hustrulid et al. 1989). The inward movement of the

walls may then create a condition closer to thepassive pressure case.

0

0.4

0.8

1.2

0 2 4 6x (m)

yy

(MPa)


overburden stresses

M arston theory

K = 3

K = 1/3

K = 1/2

a)

0

0.1

0.2

0.3

0 2 4 6x (m)

xx

(M

Pa)


overburden stresses

Marston theory

K = 3K = 1/2

K = 1/3

b)

Figure 7: Stresses calculated at the bottom of the vertical stope,

with the analytical and numerical solutions; a) vertical stress

yy; b) horizontal stress xx.


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v h= 2v

H=45m

depth = 250 m

0.5m

x

y

B= 6 m

60

h

void space

backfill

rock mass

rock mass

stope

Figure 8. The inclined backfilled stope modeled with FLAC-

2D (properties are given in Fig. 3).

0

0.1

0.2

0 9 18 27 36 45

h (m)

xx

(MPa)

mode ling with FLAC -2 D

overburden

Marston theory

K = 1/2

K = 3

K = 1/3

a)

0

0.2

0.4

0 9 18 27 36 45h (m)

yy

(M

Pa)

modeling without FLAC-2D

overburden

M arston theory

K = 1/2

K = 3

K = 1 /3

b)

Figure 9. Comparison between stresses obtained with

numerical and analytical solutions along the central line of the

inclined stope: a) horizontal stress xx; b) vertical stress yy.

When a stope is excavated in a single step, wallconvergence essentially takes place before any

backfilling. If the rock mass creep deformation is

negligible, the numerical modeling approachpresented here is more appropriate. In this case, the

Marston theory, with the "at rest" reaction

coefficient (K = K0) can be used to estimate theinduced stresses in a narrow vertical backfill (see

Figs. 5 to 7), at least for preliminary design

calculations.

4.2 Marston theory limitations

Analytical solutions can be useful engineeringtools as they are generally quick, direct and

economic when compared to numerical methods.

However, analytical solutions are only available forrelatively simple situations and may involve strong

simplifying hypotheses. For instance, with theMarston theory, the shear stress along the interface

between the rock and fill is deduced from theCoulomb criterion (see details in Aubertin et al.

2003). Its value then corresponds to the maximum

stress sustained by the fill material, as postulated inthe limit analysis approach (e.g., Chen & Liu 1990).

However, numerical simulations indicate that this

assumption is not fully applicable. Figure 10 showsthat for the vertical stope analyzed here the

maximum shear stress is only reached near the

bottom part of the stope. Hence, arching effect andstress redistribution are thus exaggerated.

Another important assumption in the Marston

theory is that both the horizontal and vertical

stresses are uniformly distributed across the fullwidth of the stope. Results shown in Figure 11

indicate that this is in accordance with numerical

calculations for the horizontal stress component(Fig. 11a), but not for the vertical stress which

shows a less uniform distribution (Fig. 11b). Also,

this simplified theory considers that the reaction

coefficient, K, depends exclusively on the fillproperty and not on the position in the stope. Results

shown in Figure 12 indicate that this hypothesis is

not too far from the numerical results. Near theboundary, the value of K would nevertheless bebetter described by aKvalue betweenKaandK0.

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0 9 18 27 36 45

h (m)

xy

(MPa)


M arston theory

K = 3K = 1/3

K = 1/2

Figure 10. Comparison of shear stress distribution along thewall.


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Work is underway to modify the analytical

solution to extend the use of the Marston theory to

more general cases.

a)

0.04

0.06

0.08

0.1

0.12

0 2 4 6x (m)

xx(

MPa)


at 1/4H

at 3/4H

at 1/2H

b)

0

0.1

0.2

0.3

0 2 4 6x (m)

yy

(MPa)


at 1/4H

at 3/4H

at 1/2H

Figure 11. Distribution of lateral pressure xx a) and vertical

stress yyb) obtained with FLAC-2D across the full width atdifferent elevations of the vertical stope.

4.3 Constitutive behavior

The reliability of any numerical calculations

depends, to a large extent, on the representativity of

the constitutive models used for the differentmaterials (and on the corresponding parameters

values). In this paper, a Coulomb plasticity model

(see Fig. 3) was employed for the fill material. Thismodel is representative of some aspects of the

mechanical behavior of backfill, such as thenonlinear relationship between the stress and strain

(e.g., Belem et al. 2000, 2002). However, this

simplified model neglects some importantcharacteristics of the media, including its pressure

dependent behavior under relatively large mean

stresses. More representative models, such as the

modified Cam-Clay model, are built in FLAC (e.g.,Detournay & Hart, 1999). However, the application

of such model is not straightforward because of the

difficulties in obtaining the relevant material

parameters. The influence of cohesion due to

cementation and possible oxidation of the fill

material may also be relevant to include in the

analyses.An interesting aspect of FLAC is that it allows

user-defined models, which can be introduced with

the language FISH. The authors are now working on

introducing in FLAC a multiaxial, porositydependent criterion (Aubertin et al. 2000, Li &

Aubertin 2003) for the yielding and failure

conditions of geomaterials. This aspect will be

presented in upcoming publications.

4.4 Interface elements along the walls

As was done with a finite element code in a

previous investigation (Aubertin et al. 2003), some

calculations were also performed with interfacesincluded in FLAC-2D, to represent the contact

between backfill and rock mass.

Preliminary results (not shown here) indicate thatthe presence of interfaces along the walls and floor

of the stope, which allow localized shear

displacements, has relatively little influence on thestress distribution in the stope and at its boundary.Some differences between the cases shown here and

models with interfaces nevertheless appear near the

bottom and top of the stope where some stressreorientation and concentration seem to take place.

This aspect however requires further investigation,

as the representativity of the (Coulomb) strengthcriterion and the numerical stability of the

calculations along these elements need more study.

0.1

0.3

0.5

0 2 4 6x (m)

K

at f loor

at 1/2Hat 1/2H

at 3/4H

modeling with FLAC

at rest

active

Figure 12. Reaction coefficient Kobtained with analytical andnumerical solutions across the full width of the vertical stope atdifferent elevations h.

5 CONCLUSION

In this paper, numerical simulations have been

performed with FLAC-2D for a vertical and an

inclined backfilled stope geometry. The results arecompared to the Marston theory solutions. It is

shown that the results obtained with the Marston

theory can be considered as acceptable, especially


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for preliminary calculations. Nevertheless, the

numerical results also reveal that the Marston theory

tends to overestimate arching effect, and thusunderestimate the stress magnitude near the bottom

of backfilled stope. Also, the influence of the mining

sequence can not be introduced in the Marston

theory. The numerical works shown here and in acompanion paper (Aubertin et al. 2003) demonstrate

that the filling sequence can significantly influence

the stress distribution in and around filled stopes.

For inclined stopes, the Marston theory is of limiteduse to estimate the stress magnitude. Additional

work is underway into both analytical and numericalsolutions to better describe the behavior of

backfilled stope. More work is also needed to study

the rock-fill interface behavior and the actual field

response of backfill in stopes.Other important issues also remain to be resolved,

including the possible degradation of the arch due to

low pressure (and tensile stresses), the influence of

water flow and distribution in backfilled stopes, the

evolving properties of the fill material (especiallywhen sulfide materials are present), the dynamic

response of the backfill, and the forces generated on

retaining structures.

ACKNOWLEDGEMENT

Part of this work has been financed through grants

from IRSST and from an NSERC Industrial Chair

(http://www.polymtl.ca/enviro-geremi/).

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AUBERTIN_Modeling arching effects in narrow backfilled stopes with FLAC.pdf