Analisis Estadistico de Frcturas Mit 1984

8/10/2019 Analisis Estadistico de Frcturas Mit 1984

1/40

ST TISTIC L N LYSIS

OF

ROCK M SS

FR CTURING

Gregory B. Baecher

l

STR CT

Over the pa s t ten years c o ns id e ra b le e m p ir ic a l

work

has

been

repor ted

on

th e

s tochas t ic

des-

c r i p t i on o f

rock mass

f r a c tu r ing and on

the

s t a t i s t i c a l design

o f

j o i n t surveys This

has led to

cons i s ten t

conc lus ions cn the

d i s t r i b u t i o n a l p ro pe rt ie s o f such di scon-

t i nu i t i e s

and i s beginning to lead to

improved

survey

des igns

Two

o f

the s t ronge s t conclu-

s ions appear to

be

the Exponent ia l i ty o f

th e d i s t r ibu t ion

o f spacings between

d i s -

con t i nu i t i e s

when

measured

by

t h e i r

i n t e r sec -

t i ons with sampling l i nes and the lognormal i ty

o f

d i s c on t inu i ty

t race leng ths as observed in

outcrops Cons is tent

conclus ions on

th e

form

o f

o r ie n t at io n d i st ri b u ti o n s

appear

more

e lus ive

Sampling b iases in j o i n t surveys now

seem more

pervas ive than was e a r l i e r

thought

In

addi t ion

to the

wel l

known o r ie n ta tio n b ia s in

sampling

from two dimensional outc rops propor t iona l

leng th

b ia s

in

which

la rg e r d i sc o n ti n ui ti es a re

sampled with increased proba b i l i t y

and censor ing

b iases in which l a rge r

d i scon t i nu i t i e s

a re

of ten

only

p a r t i a l l y observed complicate s t t ~ s t i l

infer .ences . These r e su l t s a re reviewed aga ins t

a

r e c e n t s tudy

involving some

15 000 da ta

1

Associa te

Professor of Civ i l Engineering

Massachuset ts

In s t i t u t e of Technology Cambridge 02139


2/40

INTRODU TION

J o i n t s u r v e y s a re

a n

i n t e g ra l component o f s i t e charac te r i za t ion

s tud ies in

r o c k

engineer ing

b e c a u s e

the s t r e ng th

d e f o r m a t i o n

a n d flo w b e h a v i o r o f r o c k m a s s e s a re s tro ng ly inf luenced by

th e g e o m e t r y and

engineer ing

proper t i e s o f r o c k

mass

d i scon t in -

u i t i e s F o r many d e c a d e s th e c o l l e c t i on o f g e o m e t r i c data

on

r o c k

mass

j o in t i ng h a s b e e n

r e c o g n i z e d

a s

a

p r o b l e m

o f

s t a t i s -

t i c a l s am p l i n g and

begining

in t h e m i d - 1 9 6 s many w o r k e r s

have

d e v o t e d e f f o r t to d e v e l o p i n g

sound

s u r v e y

p r o c e d u r e s a nd

to i n t e r -

pre t ing

the empi r ica l

da ta

base

The

p u r p o s e o f

t h i s

p a p e r i s to summarize empi r ica l

r e s u l t s

w i t h sp e c i a l re fe re nce to

work

a t

MIT

[6]

an d to

d iscuss

th e

in f luence

o f presen t

c o n c e p t s o f

j o in t i ng geometry

on

s a m p l i n g

p r o c e d u r e s .

G OM TRY O JOINTING

In t h i s sec t ion th e

g e o m e t r i c

p ro pe rt ie s o f j o in t i ng obser -

v a b l e

in cornmon s u r v e y s

are

d i sc r ibed W h i l e these

g e o m e t r i c

proper t i e s seem na tura l ly to f a l l

in to

d i s t i n c t

g e o m e t r i c

c l a s s e s

in

r e a l i t y

t h e y

a re

o n l y

face ts

o f o ther more f u n d a m e n t a l ways

o f d es crib in g

j o i n t

g e o m e t r y .

I t

i s t he r e fo r e im po r t a n t when

i n t e rp r e t i ng t h e impl ica t ions o f s u r v e y r e s u l t s

fo r

pred ic t ing

a g g r e g a t e

r o c k mass

behavior

t h a t these o b se rv ed g eo m et ric pro-

pe r t i e s b e viewed

as

s t rongly in terdependent


3/40

, /

In j o i n t surveys ,

th ree

geometr ic proper t i e s a re commonly

o f i n t e r e s t :

Densi ty

e . g . ,

spac ing ,

f requency , s ize

e .g . ,

t race l eng th , a rea , and

o r i en t a t i on

e .g ., s tr ik e and

dip

o f an

approximat ing plane ,

d i r ec t i on

cos ines o f the

po le .

The

measures

adopted here a re

spac ing ,

t r ace l eng th ,

and po la r d i r ec t i on

cos ines .

Spacing

measured

by

the

separ ta ion o f th e i n t e r s ec t ions o f

adjacent j o i n t

t races with

a sampling l in e , e ith er

fo r in div id ua l

se ts o f

sub-para l l e l j o i n t s o r fo r a l l j o i n t s Fig . 1 . Trace

l eng th

i s

t yp ica l ly

measured

as the

l i n e a r

d is tance

between

the

end

po in t s o f th e i n t e r s ec t ion o f a j o i n t

with

an exposed su r face .

For j o in t s t h a t a re

s t rong ly

non-p lanar , o ther measures

are

some-

t imes used.

I f both

ends o f a

t r a ce a re

no t o bs er va ble , th e l ength

recorded

i s a

censored

l eng th .

EMPIRI L

T

The da ta

base fo r the

MIT

s tudy

comprised

j o i n t survey

da ta

from

seven

cons t ruct ion

and mining

s i t e s o f varying geology

Table 1 . These da ta

were p rin c ip al ly c ol le cte d f or

engineer ing

purposes ,

inc luding

foundat ion and s lope

des ign,

and al though

co l lec ted using

a

va r i e ty o f survey procedures , these procedures

were documented

so t h a t

s t a t i s t i c a l conclus ions could

be

drawn

from the

da ta s e t s . Ind iv idua l surveys recorded

spac ings , t race

l eng ths ,

or i en t a t i ons , phy sic al c on ditio ns

e . g . , ex ten t

o f

weather -

i ng , censor ing , type o f

t e rmina t ion e . g . ,

whether

a j o i n t ended

aga ins t ano ther j o i n t , e t c . , and occas iona l ly o th er f ea tu re s.


4/40

Typical

r e su l t s fo r

spacing

d i s t r i bu t ions

are shown in

Fig . 2,

p lo t t ed

aga ins t

exponen t i a l cumula t ive

dens i ty

funct ions

cd f ,

F s

= exp{-As}. For 8

o f

the spacing d i s t r i bu t ions

ana lyzed

the ~ x p o n n t i l model f a i l ed

to

sa t i s fy Kolmogorov-Smirnov

c r i t e r i a fo r 5

Type

I

e r ro r .

This

seems

to

ver i fy the appl i ca

b i l i

ty

o f the exponent ia l model .

The

re , la t ion between

sample

means and

th e s tan dard devia t ions

i s shown in

Fig .

3. In

the

e xp on en tia l c as e mean

and

s tandard

devia t ion should be

equa l .

While average spacing var ies with or i en ta t ion o f the sampling

- i ne ,

exponent ia l i ty

does not . This can be seen in Fig .

4,

which

presents examples

o f

mean

spacings and coe f f i c i en t s

o f

va r i a t ion alon non-coplanar sampling

l i n e s .

Trace length d i s t r i bu t ions

do

no t

exh ib i t

the

cons i s t en t

cha r a c t e r i s t i c s

t ha t spacings do; how ever, in 82 of

the samples ,

t r ace

l engths

s a t i s f i ed

5

goodness -of - f i t

t e s t s

fo r

lognormal i ty .

In

th e e xc ep tio na l

cases ,

the d i s t r i bu t iona l forms f a i l ed to

sa t i s fy

X

2

o r K-S t e s t s a t

5

fo r e i t he r lognormal , gamma, normal ,

o r exponent ia l d i s t r i bu t ions ; al though, l ike l ihood r a t i o t e s t s

p lace

these

e xc ep ti on al c as es

c lose r

to lognormal i ty

than to

the

o the r d is tr ib u tio n s te s t ed

Fig.

5 , and

decreas ing

the Type

I

e r ro r to

al lowed

ce r t a in

of these

d is tr ib u tio n s t o

be

accepted

Table 2 .


5/40

For a l imi ted

number

o f

case s , da ta

a re a va ila ble on t r ace

lengths observed on or thogona l

planes

F ig . 6 ) .

For

most o f

t hese ,

length

p df s e xh ib it

littl

di f fe rence

between

s t r i ke

and apparent d ip d i r ec t ions , when i nd iv idua l j o in t s se t s are

considered separa te ly . When data

a re

no t

separa ted

by j o i n t

s e t ,

the or i en t a t i on of the sampled face

inf luences the

r e l a t i ve pro

por t ions

of

jo in ts

from

i f f r ~ n t

se t s be ing sampled,

and

thus the

t r ace length

d i s t r i bu t i on s .

Much l e s s success was enjoyed

in

f i t t i ng ana ly t i ca l forms

to or i en t a t i on d i s t r i bu t i on s .

In

a l l , 22 data se t s were ana lyzed,

and each of th e fo llowin g d is tr ib u ti on s t es te d by X

and l i k e l i -

hood r a t i o

methods: Fishe r , b iv a ri at e F is he r, Bingham, bivar -

i a t e normal , and Uniform. Data were sor ted

in to

c lu s t e r s and

maximum l ike l ihood es t ima tes

made of

d i s t r i bu t i on parameters .

Resul t s are shown in Table

3.

For many data se t s no

ana ly t i ca l

form

provided a sa t i s f ac to ry

it

based on X

Based on log

l i ke l ihood r a t i o s , th e

Bingham and b iva r i a t e

Fishe r

appear

to

pro vid e the

be t t e r f i t s .

IMPLICATIONS EMPIRICAL FINDINGS

The

empi r i ca l

f ind ings fo r j o i n t spac ing

and t r ace length

are

im ila r to

those

repor ted

elsewhere

in

the

l i t e r a t u r e

Table

4 ) ,

al though perhaps more broadly ve r i f i ed . The common observat ion

t h a t

j o i n t spacing a re e x po n en ti al ly d i s t r ibu ted along

any

l i ne t rough

the rock

mass

and

t h a t

a ve ra ge spa ci ng s

a long non-pa ra l l e l l i ne s


6/40

fo llow simple

t r i g o n o m e t r i c r e l a t i o n s

may

be

i n t e r p r e t e d t o

imply

t h a t

j o i n t s

a r e

randomly

and

independen t ly

p o s i t i o n e d in

space

i . e .

t h e i r i n t ersec t ions

w i t h

a rb it r ar y l i n e s a r e oisson

Whether spacing

o r

o t h e r geometr ic

p r o p e r t i e s

a re a ct u a l l y

independent from one j o i n t t o th e next has y e t t o be answered

f o r t h e g e n e r a l c a s e . Var iograrns have been e s t i m a t e d f o r j o i n t

p r o p e r t i e s

[ 1 2 , 1 3 ] , b u t

s p a t i a l

c o r r e l a t i o n s

a r e

d i f f i c u l t t o

v e r i f y . Because

s p a t i a l

c o r r e l a t i o n s would

imply

e i t h e r

c l u s t e r

ing

p o s i t i v e )

o r d i s p e r s i o n n e g a t i v e ) ,

they

would

be

expected

t o modify t h e e x p o n e n t i a l i t y o f

spacing

p d f s . However t h e s e

d i f f e r e n c e s

might

be

masked by

sampling

v a r i a t i o n s .

wo

examples

o f t y p i c a l

a u t o c o r r e l a t i o n

f u n c t i o n s

f o r j o i n t s e t s e X h i b i t i n g

s p a t i a l

s t r u c t u r e a r e

shown in F i g . 7.

The

i m p l i c a t i o n

o f lognormal i ty

o f t r a c e l e n g t h s i n u n c l e a r .

Lognormal

d i s t r i b u t i o n s

o f

geometr ic

p ro p e rt i e s a re

common

o b s e r -

v a t o i n s

in

geology b u t

may

merely be

an a r t i f a c t o f

sampling

b i a s e s a s d is c us s e d below [ 2 ] . I f t h i s i s t r u e , then more r e f i n e d


7/40

s t a t i s t i c a l work o r more c r e a t i v e d a t a c o l l e c t i o n schemes

w i l l

be r e q u i r e d t o

c h a r a c t e r i z e j o i n t s i z e

d i s t r i b u t i o n s .

one i m p l i c a t i o n o f

a n a l y t i c a l

forms seldom

p r o v i d i n g

s a t

i s f a c t o r y

approximat ions t o

o r i e n t a t i o n

d a t a

i s t h a t a t t e n t i o n

p ro ba bly s ho uld

be swi tched t o n o n - p a r a m e t r i c a n a l y s i s .

MO LS OF JOINTING

To d ev el op s am p lin g

p l a n s and

i n t e r p r e t

t h e i r

r e s u l t s , a

concept

o r

model

o f

t h e

geometry

o f

j o i n t i n g

i s

needed.

I d e a l l y ,

such

a model would

be

s p e c i f i e d by a l i m i t e d number o f

p a r a m e t e r s ,

and

be

s imple

enough t o be i d e n t i f i e d

from

normal f i e l d o b s e r

v a t i o n s .

S e v e r a l models

o f

f r a c t u r e geometry

have

been

proposed

i n

v a r i o u s l i t e r a t u r e s , b u t t o

t h e a u t h o r s

knowledge only two

have

found

use

i n

j o i n t

s u r v e y s .

These

a r e

t h e

random disk

model

] and th e Poisson

f l a t

model [19] . The f i r s t has

t h e

b e n e f i t

o f f l e x i b i l i t y

and

perhaps r e a l i s m ; t h e second h a s

t h e

b e n e f i t

o f mathemat ical t r a c t a b i l i t y and power. N e i t h e r , however

i s

p r e d i c a t e d on

a

m e c h a n i s t i c

concept o f

j o i n t

development.

The r a n d o m - d i s k model i d e a l i z e s

j o i n t s

a s bounded p l a n a r

f e a t u r e s

o f

random

s i z e

and

o r i e n t a t i o n ,

randomly

p o s i t i o n e d

t h r e e d i m e n s i o n a l

s p a c e . The shape o f t h e s e f e a t u re s

may

be

f i x e d

e . g . ,

c i r c l e s ) o r

a l l o w e d

t o v a r y

w i t h i n r e s t r i c t e d

f a m i

l i e s . e .g . , e l l i p s e s ) . F o r

c e r t a i n

a p p l i c a t i o n s o n l y t h e assump


8/40

8

t ion

of

convexity i s

requi red

[17]

Fig

8) . The model i s

speci f ied

by

an

in tens i ty

measure

(e .g . , number

o f

j o i n t

centers

per

rock

volume ,

or ien ta t ion

d is t r ibu t ion

parameters , and s iz e d is tr ib u ti o n

parameters.

- _ 0 _

.--

.

_.0

...

_

. __

r _ _ _

; 1

The

Poisson f l a t model idea l i ze s

j o in t s

as convex polygonal

fea tures

formed by

the in te r sec t ions o f a Poi s son- l i ne process

on

pianes which a re themselves the r ea l i z a t i ons o f

a

Poisson

piane

process

in

space

R

3

Fig

9

) .

The

shape

o f

the

fea tures

a re f ixed only to

the

ex t en t

o f be ing polygons and may have an

average ob l iqu i ty

d i f f e r en t

from 1 .0 ,

Within

a

l a rge volume

o f rock

a number o f j o i n t s can be co-p l ana r . The

model i s spec i f i ed by

a dens i ty

o f

planes

in

R

3

a

dens i ty

of

l ine s

in

R

2

o r i en t a t i on

d i s t r i bu t i on parameters fo r the

planes

.

and

l i ne s ,

and

a

co lor ing

ra t io

which randomly

ass igns

po ly -

gons as

j o in t s

o r

as

rock br idges . Within

t h i s

model both spac ing

and t r ace

l ength pd f s a re

necessar i ly

Exponent ia l .

The

importance o f

these

models

to s t a t i s t i c a l sampling

and

in fe rence

t ha t

they

provide

an

o rg an iz in g re fe re nc e w ith in

which

to

i n t e r p r e t da ta , and

a

l imi t ed

number

o f

parameters

with

which

to

summarize

i n fe rences .

.


9/40

S MPLIN PL NS

While

the design

o f

j o in t

surveys

i s

s t ra ight forward

s t a t i s t i c a l problem in prac t i c e

s t rong geomet r i c

b iases m y

be

in t roduced by sampling procedures . I f these b iases a re

no t

accounted for , survey plans m y be s tr o ng ly non -r epr e sen ta ti ve ,

with the r e su l t

t h a t

data a re Gi f f i cu l t to

i n t e r p r e t .

Thus th e

pr inc ipa l

concern

in des igning

j o i n t

survey

s t r a t eg ie s i s

recog-

niz ing sampling b iases and c orr ec tin g f or them.

This sec t ion focuses

on types

o f

sampl ing b ia se s .

Most such

b iase s are

eas i ly cor rec ted ,

if recognized .

Sampling

theory

r e su l t s , inc luding procedures , es t ima to rs , and p re cis io ns a re

summarized

in

Ref. [ ]

Spacing

and Trace Length

M ost work to da te has cons idered sampl ing inferences fo r

spacing t r ace l ength , and or ien ta t ion as mutual ly

independent

fo r an

except ion ,

see

[14] . This i s in

f ac t

no t the case ,

bu t

s impl i -

f i e s

mathemat ics .

S ta t i s t i c a l

aspec t s

o f

in fe rences

o f

j o i n t spac ing a re

f a i r ly rou t ine , and exponent ia l sampling theory i s wel l developed.

Presuming j o in t

l oca t ions

to be

random

and independent average

spacings a lo n g n o n c op la na r sampling

l i nes

a re simply

r e l a t ed ,

and

the

combined

sample can be used

to

in c re as e e stim at e prec i s ions .

Sampling

in fe re nc es fo r j o i n t t r a ce

le ng th a re

much

l ess

simple

s ince they a re

complicated

by geometr ic b iases l ead ing

to


10/40


11/40

11

w i t h i n th e

r oc k mass ( e .g . ,

u s in g rh e r an d om - di sk m o d el ), the

sample

i s ,t he re fo re , l ik e ly to b e quadra t ica l ly biased .

Th e

e f f ec t

of a l i nea r

bias

i s shown s c h e m a t i c a l l y in Fig . 1 2 .

, .

The

p robab i l i t y

o f a t r ace l ength 1 appear ing in the

s am p l e

_

i s

the produc t o f th e p ro b ab il i ty o f t appear ing

on th e o utcro p,

f (1 )d , a nd

the cond i t i ona l

p robab i l i t y o f t in te r sec t ing t h e

s a m p l i n g

l i ne t

d o e s

a p p e a r

on

th e ou tc rop ,

k i ,

f ( )d i

= k 1 , f 1 ) d l

s

9 )

in which

k i s a n o r m a l i z i n g constant . Any h igher

order bias in t ro

d u ces

th e cond i t iona l

probabi l i ty

k in , in which k e q u a l s th e

rec ip roca l o f th e n th cen t r a l

moment

of f ( i )

.

in te res t ing pr ope r t y. o f

th e

length

bias

t h a t

t serves a s a f i l t e r

t h a t

t r a n s f o r m s

many common

.

d is t r ibu t ions

f 1 )

i n to

a p p r o x i m a t e l y

l o g n o rm al

forms

[2] .

I n

the

sense o f

cornmon

goodness -o f - f i t t e s t s t hese t r a n s f o r m e d

pd f s

a re in d is tin g u is h ab le

from

l o g n o rm al

pd f s a t r e a l i s ti c

s am p l e

s i ze s .

T h i s i s d e m o n s t r a t e d in Fig . 13 i n w h i c h l i n e a r ly

biased

e x p o n e n t i a l an d

l o g n o rm al

f 1)

I

S a re t e s ted aga ins t

be s t

t

lo gn orm als an d

shown

to s a t i s fy

K-S c r i t e r i a a t

th e 5

l e ve l .

S i n c e s ize

b i ases

a re common i n geo log ica l s a m p l i n g [ I I ] , t i s

in te re s t ing to sp ec ula te t h a t th e common obse rva t ion

o f l o g No rm al

pd f s fo r g e o m e t r i c prope r t i e s

i s

pr imar i l y

a n

a r t i f a c t o f

s a m p l i n g procedures .


12/40

12

The t r a ce l ength

da ta

of Fig . 14

were

col lec ted as

area

samples i . e .

every j o i n t

with in a very

l a rge

sampling f i e ld

was measured)

a t

the ground

sur face

(litop

of

rock ) and on the

f lo o r o f a 2 m

(60 ) deep

e ~ c a v a t i o n ( bot tom

The

j o i n t

popula t ions

a re fo r

presen t

purposes

e s se n ti al ly i d en t ic a l

and y e t the

bottom

o f rock sample has

a

somewhat lower mode and

a

much

t h i nne r

upper

t a i l .

The

reason

i s

t h a t

many

of

th e

t races

observed

in

the

excavat ion

run

o ff

in to the rock wal l s

and can

no t b e o bse rve d in

t h e i r

en t i r e t y . Since

t h i s censor ing

occurs

with propor t iona l ly higher

p ro ba bi l i ty to

longer

t r a c e s

the

sample

i s

biased toward

shor t e r

l engths and the extreme

upper

tail disappea rs

comple te ly .

Censoring

i s a wel l

known

s amp ling p roblem in l i f e t e s t ing

and

other

f i e l d s

of

s t a t i s t i c s .

For

th e t r a d i t i o n ~ l problem in

which th e po in t

of

censor ing i s

cons tan t i . e .

a l l

t r aces

longer

than

1

c

a re censored and a l l shor t e r than

c

a re observed

com-

p l e t ~ l y

a

l a rge l i t e r a t u r e

of

both

f r equen t i s t and

Bayesian

methods has been d eveloped .

Pr imar i ly t h i s

l i t e r a t u r e

deal s

wi th Exponent ial d i s t r ib u t i o n s [ 3,

J b u t r e s u l t s a lso

e x i s t

fo r o th e r forms

[ e . g . 8 9] The q ue st io n o f f ix ed p oin t

censor ing fo r

j o i n t surveys has been cons ide red

by

Cruden [ ~ J

Baecher and Lanney

[2

J and

P r i e s t

and Hudson [15J.


13/40

13

Unless

th e

sampl ing program

f o r j o i n t surveys s

c o n s t r a i n e d

such

t h a t

j o i n t s

l o n g e r

than

a

f i x ed l e ng th

1

c

a r e

n o t

measured

even they i n

f a c t

could

b e , t h e

problem o f censor ing becomes

more

d i f f i c u l t .

I n

p a r t i c u l a r , t h e p o i n t

o f

censor ing s i t s e l f

a

random v a r i a b l e . The o b s e r v a t i o n s recorded

a r e

1

a

s e t

o f

complete ly

observable t r a c e s ,

=

{1

x

, 1 , . , 1

x

, r } : and 2

a s e t

o

t r a c e s f o r

which

only one o r

n e i t h e r end

i s

o b s e r v a b l e , t

=

- z

{ l

Z

, 1 , , 1

z

,t}

The

l i k el i h oo d o f

( - ,I t s

CD

t

n f 1 . I e )

n

f 1 Jle d1

1 x ~

1

z,

~

Rz,j

10

i n

which

= th e parameters o f t h e

t r a c e l e n g t h

pdf c o r r e c t e d

f o r

o t h e r b i a s e s ) . The

secon d term i n t h e r i g h t hand s i d e i s

t h e

p r o b a b i l i t y t h a t a

censored t r a c e

would be lo ng er than t h a t

observed. C l e a r l y ,

c l o s e d form maximizat ion o f Eq. 10 with r e s -

p e a t t o

e s only p o s s i b l e f o r p d f s having a n a l y t i c a l cumulat ive

p r q b a b i l i t y d i s t r i b u t i o n s .

Truncat ion

b i a s

I n c o l l e c t i n g

j o i n t

d a t a

a d e c i s i o n s

u s u a l l y made n o t t o

record

t r a c e s

s h o r t e r

than

some

c u t - o f f

l e n g t h .

This

d e c i s i o n

i s

made

e i t h e r out

o f expediency

o r

because s h o r t

t r a c e s a r e

d i f f i

c u l t t o d i s t i n g u i s h , a s f o r example i n photographs.

S e v e r a l

w orkers ha ve

noted

t h a t

t h i s form o f t r u n c at i o n in tr o du ce s b ia s

i n t o the sampling p l a n , i n c r e a s i n g the

sample.mean [ 2 , 5 , 1 5 ] .


14/40

14

F i g . 15

shows

th e b i a s i n th e sample mean r e s u l t i n g

from

t r u n c a t i n g

a t

a

given

f r a c t i o n o f

th e

mean t r a c e

le n g t h , f o r

an

e x p o n e n t i a l

pdf o f

t r a c e

l e n g t h . This

b i a s

s m a l l e r f o r

d i s t r i b u t i o n s , l i k e th e lognormal with zero d e n s i t y a t the

o r i g i n . The f i g u r e shows t h a t th e e f f e c t o f

t r u n c a t i o n ~ s

on

e sti m a te s o f c e n t r a l tendency o f the t r a c e

l e n g t h

pdf

i s

s m a l l ,

u n l e s s t h e chosen

t r u n c a t i o n

l e v e l

i s

l a r g e e . g . ,

>1 o f t h e

mean . In

most

c a s e s

t h i s b i a s may be s a f e l y i g n o r e d .

O r i e n t a t i o n

Geometric

b i a s e s

i n j o i n t s u r v e y s , s p e c i f i c a l l y f o r

j o i n t

o r i e n t a t i o n , were brought

t o t h e a t t e n t i o n o f the e n g i n e e r i n g

l i t e r a t u r e by R. Ter zagh i [ 18 ] a lt ho ugh Sander e t a l . [16] and

o t h e r s had e a r l i e r c o n s i d e r e d

r e l a t e d

problems

w i t h t h in s ec ti o n s.

The

problem Terzaghi p oin te d o ut i s t h a t j o i n t s more o r l e s s p a r -

a l l e l t o an outcrop

a r e

sampled with p r o b a b i l i t y approaching

zero

i . e . , t h e r e i s

a

b l i n d zone

f o r any

p a r t i c u l a r

outcrop

o r

b o r i n g ) .

Since

o u t c r o p s

may

form

along j o i n t

s u r f a c e s , t h e r e i s

o f t e n a

s t r o n g p o s s i b i l i t y

t h a t an

e n t i r e s e t

o f j o i n t s i s being

s y s t e m a t i c a l l y

under r e p r e s e n t e d

i n the survey r e s u l t s .

The

c or r e ct i o n f o r

t h i s

b i a s

i s

s imple .

O r ie n ta tio n d ata

can be weighted

i n

i n v e r s e p r o p o r t i o n

t o t h e i r

p r o b a b i l i t y o f

appearing

i n

t h e

sampled

p o p u l a t i o n . Con sid er in g only

outcrop

sampling

th e p r o b a b i l i t y

o f a j o i n t

o f

s p e c if ic o r ie n ta ti o n

i n t e r s e c t i n g an

outcrop

can be seen from F i g . 1 6 t o be p r o p o r t i o n a l

~ . .


15/40

5

to

L s in a

d

Thus observed da ta should be ad jus ted by th e weight ing f a c t o r

W

a

s in

a

Terzahgi

gives

t h i s

weight ing func t ion

on equa l areas

ne t s .

For

sampling from

mul t ip le

o utcro ps th e weigh t ing func t ion

fo r some

or i en t a t i on J { A ~ V }

becomes p ropo r t i ona l to

in which B

i

i s the dimension o f outcrop Qi={ m n} i s the po le to

the be s t f i t t i n g plane to

th e o utc ro p

and Jn

= the

do t p ro

duc t .

CONCLUSIONS

The r e su l t s

of th e p re s en t ana ly s i s o f

j o i n t

survey

da ta

appear

to

be c on sis te nt in

impor tan t

ways

wi th r e su l t s pre sen ted

e l sewhere

in th e l i t e r a t u r e and in

c on ju n ct io n w i th

those o the r

r e su l

t s

appear to

j u s t i f y th ree

emp ir ic a l c oncl us ion s

on

the


16/40

16

d i s t r i bu t i ona l

proper t i e s

o f

o on j o i n t

survey measurements

Spec i f ica l ly

j o i n t sp ac in gs as measured a long sampling l i ne s

o r in bor ings

are

very

f requent ly

d i s t r i bu t ed exponent ia l lYi

t r ace len gth s as

measured in outcorps a re

of t en d i s t r i bu t e d

lognorrnallYi

and

p lan a r

or i e n t a t i ons o f j o i n t s

do

no t

appear

wel l modelled

by

common a na ly t i c a l forms

S t a t i s t i c a l

analys is o f j o i n t survey

procedures both

in

th e presen t s tudy and

as

repor ted e lsewhere in th e l i t e r a t u r e

lead

to

th e

conclusion

t h a t

g eometr ic s amplin g

b iases

a re

o on

in t r ad it io n al sampling plans and

must

be guarded

aga ins t . The

most f requen t of these a re th e w ell known

or i en t a t i on

b ia s i n

which j o i n t s s ub pa ra lle l to an

outcrop

a re unde r rep re sen ted in

samples co l l ec ted

on

the outc roPi s ize b i as i n which l a rge r

j o i n t s

a re

sampled with g r ea te r p r ob a b il it y than sma l l e r j o i n t s

are i

and

censoring

b i a s

in

which

l a rge r

j o in t s

a re

more

f requent ly

masked by overburden o r

excavat ion l imi t s than

smaller j o in t s

a re .

The impl ica t ions o f these conclus ions fo r the

des ign

o f

j o i n t surveys and fo r th e use o f survey data in engineer ing models

o f rock

masses

a re aga in o f

two types .

F i r s t c og en t r ea so n

ex i s t s

fo r adopt ing c e r t a i n

d i s t r i b u t i o n a l forms when

deve loping

s tochas t i c

models

o f f rac tu red media fo r

s t r e ng th

deformat ion

and flow ana lys i s and when designing survey procedures . Second

th e

sampling

theory

o f

j o i n t

surveys i s not

s t r a i g h t

forward and

na ive s t a t i s t i c a l es t imato r s

may

be s t rong ly

b iased .


17/40

17

KNOWLEDGMENTS

This

work was sponsored by

the

U S Bureau o f Mines

under

Cont rac t

J 27S lS

The

au thor wishes the th an k N ic ho la s A

Lanney

and W ill iam D ers ho w it z who con t r ibu ted

s ub sta n tia lly to

the

ana lys i s

o f l ength spac ing and or i e n t a t i on respec t ive ly . The

work was

j o in t l y

d i rec ted

by Herber t

Ein s t e in and h is

comments and suggest ions are gra te fu l ly

acknowledged


18/40

REFERENCES

Baecher,

G.B., N.A. Lanney, and H.R. Einstein, 1977, Sta t is t ica l

description of rock f racturs and

sampling,

18th u.S. Symposium on

Rock

Mechanics.

2.

Baecher,

G.B. and N.A. Lanney, 1978, Trace length biases

in

jo int

sur

veys,

19th

u.S. Symposium on Rock Mechanics.

3. Bartholomew D.J. , 1957,

A

problem

in

l i fe testing, Journal American

Stat is t ical

Association, 52:350-55.

4. Barton,

C.M. 1978, Analysis of jo in t

t races,

19th u.S. Symposium on

Rock Mechanics.

5.

Cruden, D.M.

1977, Describing the size of discontinui t ies, International

Journal

of Rock Mechanics and Mining Science,

14 :

133-37.

6. Einstein, H.H., G.B. Baecher and D

Veneziano, 1978,

Risk

analysis

of

rock

slopes in

open p it mines, Final

report

to the U.S. Bureau of Mines,

Contract J02750l5.

7. Epstein, B., 1954,

Truncated l i fe

tests

in

the exponential

case,

Annals

of Mathematical Stat is t ics 25: 555 f f .

8. Fisher, R.A.,

1931, The truncated Normal

distribution, British Associa

tion for

the

Advancement Science, Mathematical

Tables,

I: XXXIII-XXXIV.

9.

Hald,

A., 1949,

Maximum

likelihood

estimators

of

Normal distr ibut ion truncated a t a known point ,

32: 119.

the parameters of an

Skand. Aktuar Tidsk,

10. Jewell,

W.S., 1977 , Bayesian

l i fe test ing using th e to ta l

Q on

tes t in

Tsokos, C.P. and I.N.

Shimi

Eds.) , Theory

and

Application

Reliabi l i ty,

v. 1.

11.

Kaufman G.M. 1963,

Stat is t ical

Decision and

Delated

Techniques

in

Oil

and Gas

Exploration,

Prentice-Hall.

12. LaPointe,

P.R., 1980, Analysis of

th e

spatial

variation

in

rock

mass

properties through geostat is t ics , 21st u.s . Symposium on Rock Mechanics.

13

.Miller,

S.M., 1979 ,

Geostatis t ical

analysis

for eva luat ing

spatial

dependence in f ra ctu re se t characteris t ics , thesis submitted in parial

fulfillment

of the

requirements

for

the degree Master

of

Science,

Univer

sity of Arizona.


19/40

14.

Pahl P J

1981

Estimating th e mean

len gth o f discontinuity

t races

International Journal o f Rock Mechanics and Mining

Sciences

18: 221-228.

15.

Priest

S D

and

J .

Hudson

1981

Estimation

of d iscont inui ty

spacing

and trace

length

using scan1ine surveys International

Journa1.2i

Rock

Mechanics and

Mining Science 18:183-199.

16. Sander B. 1926 Zur petrographisch-tektonisschen Analyse I I I Jahrb.

d.

geol.

Bundesanst. Wien

17 . Santalo L. 1976

Stochastic

Geometry and Integral Calculus Addison-Wesley.

18. Terzaghi R. 1964 Sources

of

errors in

jo in t surveys

eo technique

15 : 287-304/

19.

Veneziano

D.

1978

probabilistic

model of joints

in

rock M T working

paper.


20/40

F i g u r e Number

1

2 .

3 .

4 .

5.

6 .

7

8 .

9 .

1 0 .

FIGURE TITLES

Ti t le

T yp ic al c ha lk

sampling

l ine

on

an

o u t c r o p ,

showing def in i t ions

o f jo in t

sp acin g

A

and t race

le n g th 1 ).

Jo in t

sp acin g frequency dis t r ibu t ions f o r

s ix 6)

s i t e s Sample s izes between and

2000; plot ted

d a t a fo r f i f teen in tervals o f

each

d a t a

se t

Mean sp acin g s and s t a n d a r d

d e v i a t i o n s

o f

sp acin g f o r in te rsec t ions

o f

jo in ts

w i t h

sampling

l i nes D i f f e r e n t symbols refer to di f ferent s i t e s

F or


dis t r ibu t ions

th e mean

and

s t a n d a r d

d e v i a t i o n

s hould be th e same, as

shown

by

45

l i ne

Mean sp acin g s among jo ints and coeff ic ients

of

varia t ions

stan d ard deviation/mean) f o r

in te rsec t ions sampled

a lo ng n on -c op la na r

sampling l ines w ith in th e sarne rock mass. F or


dis t r ibu t ions

th e

Cov s hould eq u al

1 0

Cumulative

dis t r ibu t ions

o f j o in t

t race l e n g t h

fo r one s i t e having tw o d i s t i nc t s u b p a r a l l e l

se ts

Trace

l e n g t h s

measured

in

h o r i z o n t a l

o u tcro p s

i . e .

s

t r ike

and

in

ver t i ca l o u tcro p s

approximately

para l l e l

to d i p s i . e . IIdi

ps

l l

B e s t f i t t ing cumulative dis t r ibu t ion f u n c t i o n s

fo r

s t r ike

and

d ip

l e n g t h s

o f jo in t

t races

a t

same

s i t e

A utocovariance f u n c t i o n s f o r

jo in t

d i p , showing

l i m i t e d alth o u g h def in i te

spat ia l

corre la t ions

Random

d i s k

model fo r

j o in t s

P ois s on

f l a t model

f o r

jo in t s

Prof i le view o f j oi nt s i nt er se ct in g

an

o u t c r o p .

Note

tha t

larger jo in ts have a p r o p o r t i o n a t e l y

g r ea te r p robab il it y

o f s t r ik ing

o u tcro p than do

s m a l l

j o in t s


21/40

Table

ata ase fo r Empir ical

o i n t

S t a t i s t i c s

SITE

PURPOSE

GEOLOGY

GREEN COUNTY

NUCLEAR

POWER FOLDED SEDIMENTA

SITE

A

NUCLEAR

POWER

HIGH

GRADE

t ETAMORPH

ICS

SITE B

NUCLEAR

POWER

SHALLOW

WATER

SEDIMENTS

BLUE HILLS

STUDY

AREA GRANITE PORPHYRY

AND

VOLCANICS

PINE

HILL

STUDY

AREA GRANITE

AND

VOLCANICS

DUVAL

MINE

COPPER PORPHYRY

PINCOCK ALLEN

MINES VARIOUS

MAINLY

AND HOLT--VARIOUS

COPPER

PORPHYRI

SITES


22/40

Table

2

Resul t s of Chi Square Goodness o f f i t

Tests fo r

Trace

Length

is t r ibu t ions

SITE

EXPONENTIAL

MM LO NORM L

Site A

top fail

fail

fail

Site

A

bottom

fail

fail

fail

Site A

sides

fail

fai l

pass

Site

A

sides

fail

fail

pass

Greene

Co.

fail

fai l .

pass

trench A

Greene Co.

fail

fai

1 pass

trench B

Greene

Co.

fail

fail

pass

trench

C

Greene

Co.

fail pass

pass

trench T

Site

fa il fa il pass

Blue

Hills fail

fai l pass

Pine

Hills

fail fail pass

all

significance

levels

set

at 5

except

where

in-

dicated

by

*) which were

set at

1 .


23/40

Table 3

Goodness o f t r e su l t s

fo r

or i en t a t i on da ta

JOINT

SET

Fisher

ivar ia te

Fisher

Bingham

x

2

t es t

A

558

420

480 none

B 80

8

none

lC

294

144

107 none

A

127

32 47 none

B

72

60

282 none

C 442

323 283

none

D

131

91

326

none

2E 29

none

2F

89

69

51 none

3A

125

122

114

a l l

3B

20 none

3C

20 20 91

none

4 568

555 517 none

SA

445 288

272

none

5B

236

142

149 Bingham

6A

79

57

37

none

6B

62

27 27

a l l

7A 383

298 280

none

7B

251

91 140 none

7C

574 575 555

none

7D

120 45

41

Bingham

ivar ia te

Fisher

8 295

144

107

none

Tota l

Bes t Fi t s

13

d i s t r i bu t i ona l

forms s a t i s fy ing

goodness

o f t

c r i t e r i on

at

5

confidence.

~


24/40

Table 4 Best

f i t t ing

dis t r ibu t ions fo r

jo in t

spacing

and t race length, as

reported

in the

l i t e ra tu re .

Fi t t ing procedure and goodness-of-f i t t es t ing vary

from

one

source to

another.

Predominant rock

mass

geologies also vary.

SOUR E

SP ING TR E

LENGTH

OLO Y

Barton 1977)

Bridges 1976)

Call, et a1. 1976)

Cruden 1977)

McMahon 1974)

Priest Hudson 1977)

Robertson 1970)

Snow l970)

Steff-an, e t a1. 1975)-

Exp

Exp

Exp

logN

logN

Exp

Exp

logN

Exp

Exp

-Metamorphic

Metamorphic

Copper Porphyry

Various

Various

Chalk and

various

DeBeers

Mine

Crysta11ines

~ t m o r p h i c ~

and

various


25/40

t


26/40

en

o

X

o

g

ro J

Z J

we:

en

Q

Z

o

:

_

cn

ro

roO:

zo

Q.

o

Q J

~ e :

ic E

n

o

t

o

v

o

N

o

o

A J . 1 1 1 8 1 8 0 ~ d 3 A I J . 1 1 n W n ~


27/40

1

8

4

6

MEAN SPACING

2

1 .. .

2

8

Z

>

6

W

Z

4

CJ


28/40


29/40

.

.

.

5.0475

1

.5

8

9

8

7

>

60

STRIKE SET 2

.J

50

m


30/40

-

t::

o

n

m

W

o

IC >

:E

,

Q

o

Q

.L:I H LE>N3 : : > V ~ l


31/40

5

4

3

laJ

2

Z

1

2

5 10

SEPARATION M


32/40


33/40

O NT

~ ~ ~ S U R F C E


34/40


35/40


36/40

r

P F

SAMP L E

TR E LENGTH


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99

BIASED

logNormal

KOLMOGOROV SMIRNOV

BOUNDS. 5 n=100

\

BEST

FIT

logNormal

95

1

9

8

t

d

7

m

ti

30

20

o

4 5 6 7 8 9 1 20

TRACE LENGTH FT)

30

40 50 60


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....

100

_

:

:

C)

1

J

Z

o

SITE A

TOP

SITE A:

BOTTOM

1 0 L _ L L

JL

...L.

..L... _

1

1

50 90 99

100

CUMULATIVE FREQUENCY )


39/40

LaJ

E

LaJ

:

LaJ

25 5 75 1

TRUNC TION LENGTH ME N


40/40

~ ~

Analisis Estadistico de Frcturas Mit 1984

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Transcript of Analisis Estadistico de Frcturas Mit 1984