A M E T H O D FOR D E T E R M I N I N G T H E T O T A L S T R E S S E S

I N S O L I D R O C K F R O M T H E D I S T U R B A N C E T O T H E S T R E S S

F I E L D NEAR A B O R E H O L E W I T H A D E F O R M E T E R

V, S . A k i m o v , V , S. K u k s i n , M, V. K u r l e n y a , a n d A , V, L e o n t ' e v

UDC 622,831

Present load re laxat ion methods for de termining the total stresses in solid rock involve dr i l l ing a rock core. and require specia l equipment to ensure its preservation.

Let us consider a method of de termining the total stresses in an isotropie e las t ic rock mass without dr i l l ing out a core. I t is essential ly as follows [1], From a mine working at a given depth we driU a measurement hole o f radius r = 1 (see Fig, 1), In this we set up a deformeter to measure the radia l d isplacements of the periphery (contour) of the hole, After taking the in i t i a l readings of the deformeter para l le l to the measurement hole a t a d is tance ! we dr i l l a hole of radius R ~ 1, As a result, the stress field around the measurement hole will a l ter , which will cause deformat ion of its periphery and changes in the deformeter readings, The required values and di rec t ion of the stresses (ol , oz, and e) are then found by ca lcula t ion ,

The problem of the state of stress of such a rock mass reduces to the plane problem of the theory of e las t ic i ty for an infini te region with two arbi t rar i ly located circular holes of different d iameters when compressive forces a t and oz ac t a t infinity, We can find two ana ly t ic functions ~(z) and X(z) of the complex var iable z which cha rac - ter ize the state of stress of the plane with its two holes, The region of the complex var iable z comprises the whole infini te plane outside the holes. The functions 9(z) and ~(z) are related to the stresses and displacements (strains) as follows:

o ~ + ~ = 4Req~' (z) ;

o ~ - - o y q - 2 i ' r ~ y : 2 [ ( ~ - - z ) rp" (z) - t - ; ( ( z ) ] ; (1)

2a ( U + i V ) = • + (~--z) cp' ( z ) - - x (z).

H e r e

~(z) = ~ ( z ) + z ~ ' ( z ) ~ (2)

R(z) is a function which is invar iant with respect to d i sp lacement of the origin along the x axis; r (z) and r are the Kolosov--Muskhelishvil i functions for complex representat ion of the displacements and stresses [2],

Let us consider the s tress-deformation state of the rock around the l a r g e - d i a m e t e r borehole, neglec t ing the

presence of the s m a l l - d i a m e t e r one, The error of a ca lcu la t ion based on this assumption can be es t imated by ver i - fying the sat isfaction of the boundary conditions on the contour of the l a r g e - d i a m e t e r borehole [3],

If a force o I acts a t an angle O to the y ax i s the s t ress-deformation s tate of the undisturbed rock mass is given by the functions

(3) (l t ~',~o ' ~ ~ e 2% ~ (z) = -~ z; (z) = -

Institute of Mining, Siberian Branch. Academy of Sciences of the USSR. Novosibirsk, Translated from Fiz iko- Tekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh. No. 1, pp, 13-18, January-February. 1974, Original a r t i c le submitted May3"0. 1973.

�9 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or trc~smitted, in any form or by any means, electronic, me- chc~ical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00.

l0

~ '+' \ I ] x . ~ N I z"-4 ~ " , �9 /////////~,

ar

Fig. 1. Determination of the displacement of the contour o f a small borehole by means of the disturbance of the stress field near it due to a large borehole.

or at +(o :) (z) = -~ z; X(o ~) (z) = ~ a~z (2e 2i~ + 1).

(4)

The functions r (z) and X[t)t (z) representing the additional stresses in the rock due to the large-diameter borehole can be found from the boundary

conditions

�9 a, (t + r e - 2 ' ~ ~f~ (t) + (t - ? ) ~I ') '(t) + z~ ') (t) = - -~ (~)

where t is the affix of the point z on the contour of the large hole. Using the method of N. I, Muskhelishvili. we find

zl ') (z) =

aiR2 e -2i0 + l ' ( z ) - 2 ~ - z '

o1R4 e-2i~ atRa (e - 2 i ~ I). 2 ( z - - t ) 3 - [ - 2 ( z - l )

(6)

(7)

The boundary condition for the functions r and X~ l) (z), which characterize the additional stress-defor-

marion state of the rock appearing after the smal l -diameter ti6rehole is drilled, will be

_ a , ( t 42 ) (0 + (t ~) +(~')'(t) + x(2 ) (t) = - -~ +

at[R" e_2, o (t_7~)R ~e2~~ R 4e2i~ RZ ] q- "2- (-/-~T~/) - - (7 - - l)2 ~- (7 - - / ) a ~r- ~ (1 - - e 2 i ~ .

From Eq, (8) we find that

(8)

1 e_2~ 0 R*e2i~ (1 - - z 2) R':' �9 R4zSe 2lo R2 ( l _ e -2 i0) ] at ~ ~ (1 - - lz) 2 (1 - - lz) a (l--/Z) (~(2 l) (z) = .~ + + V ze2'~ + 4 z ' (9)

R2 e2iOJ ~[91)(Z) : (Z__ l ) qg(21)t(Z)__O'l Z~2' (:to)

The function ~(l)(z) and X (I) (z) defining the state of stress of the rock mass with two holes, acted on by at, can be written in the form

~o ~ (z) = ~(o" (z) + ~o~ ~) (z) + ~o(~ ') (z), (11) z(') (z) = X(o')(z) + z~')(z) + ~())(z).

Let us consider the action of forces o z directed at an angle O to the x axis. The solution can be found from the previous substitution of the angle (9+ ~ / 2 for (9, Then e zi19 is replaced by

e 2 ~ e + ~ - - e 2 i~ (12)

From (9), (10), and (12), we find

a 2[ 1 e_2 i 0 R 2 ( 1 - z z) ze 2i0 ~ 2 ) ( z ) = T L ~ - - - ~ f : ~ Y - -

R 2 R%ae 2i~ R 2 (I + e 2i~ ] i~ ze2i~ ( l _ l z ) a @ (1 - - l z ) z , (13)

�9 R 2) zR z ] - - e - 2 * ~ ~ + (1 - - tz) e2ie " (14)

The functions representing the stress-deformarion state of the rock compressed by a force o z are equal to

+~ (z) = ~ ) (z) + +i~) (z) + +(2)(z), z~ (z) = 7,~ -~ (z) + zi~) (z) + x(2) (z).

(i 5)

11

Using Eqs. (11) and (15) and the method of superposition, we arr ive at expression (1) for the d isplacements of the rock mass weakened by two holes and compressed by forces o ~ and o z, in which

~(z) = ~0 ') (z) + ~') (z) + ~(~'~ (z) + ~0 ~ (2') + ~[~ (z) + ~7 ) (z), / 7,(2') = xg)(z) + x~,)(z) + ~(~,)(z) + z(g) (z) + x} ~) (2') + x(~ ~) (z). l (16)

The deformeter will record the d isplacements of the contour of the s m a l l - d i a m e t e r hole due to the dr i l l ing of the l a rge -d i ame te r one, These displacements will be

2G (U -}- iV) = • (t) -i-- (-t - - t) r (t) - - Z (t) - - • (l) - - (-t - - t) (p; (t) -}- Xa (t), (17)

where

q)3 (2') = 0"1 - L fro r O?, - - O1 e - 2 iO ' 4 " 2' - ~ " 2 z '

Z i 0 1 - - ~ z e 2 i ~ -~ q~ - - cJ 1 e - - 2 i O 1 e - 2 i 0 ~.~ (z) = 7- (Ox + ~ ) 2 2 ~ 2= (o~ + o2) ~

From (17) and (18), for the rad ia l d isplacements of the borehole periphery we get

( i s )

R2((r~--ch) { [cos2(Oq- :oQ-- lcos (20-b~ Ur : 4G • 1 - - 2l cos ~z ~- l s - -

( 1 - - l'~) cos 20 - - cos 2 (O + ~) - - 2l [cos (20 - - ~) - - cos (20 + a)] + l s cos 2 (O - - o~).

cos 20 I s (1 - - 21 cos r162 @- lS) ~

( 1 - - 21 cos o~ -}- 12) "

R~ cos 2 (0 d- oQ - 3l cos (20 -~- ~x) n L 3l s cos 20 - lZ cos (20 - o~) -t-

(~2 + a~) I - - l cos 0~ -,l- cos 20 - - l cos (20 + ~)] __ [cos 20 @ ( o 2 - - ( r l ) 1 - - ~ c o s ~ , 1' 1---2l-c~e;-~l 2 [ T ff

c o s 2 ( O - - a ) - - l c o s ( 2 0 - - a ) ] } : ] ' - - - - - " ~ ' c ~ ' ~ _ . (19)

This expression relates the radia l d isplacements of the periphery of the s m a l l - d i a m e t e r borehole to the stresses in the rock mass and takes account of the di rect ion of ac t ion of the stresses.

From the equation for the displacements (19), i t follows that to find the required value of o x, ~ and t9 (the angle de te rmin ing the di rect ion of their ac t ion re la t ive to the plane o f measurement) , i t is necessary and sufficient

to find the displacements in three arbitrary directions a I, a z, a 3. Let a l = O, az=19+ ~r/2, % = O+ ~r/4, where a i is the angle between the d i rec t ion of ac t ion of one of the principal stresses and the d i rec t ion in which d i sp lacement U x is measured. Using these values of the a i, we get the following system of equations:

U,=A (o2+o~) +B cos 20(02--a~),

U2=F (02+ol) + (C cos 20+D sin 20) (o2--~1),

(20)

(21)

Ua~--- ( o 2 - - o l ) ( K s in 2 0 + L cos 2 0 ) -{- (o2q-o l ) M, (22)

where

R 2 ~r R2121+ ] • 2 ~r q- l] A ~--- 4G 1 - - l; B = 4-d (l - - / ) i t 2 ;

R'[ 2• •247 (3t'--I) , ] C = u -~ 1+/2)2 l 2 R~'x(I + / s )3 l + l S ;

R~-[ [ 4, R s t ( 3 - - / s ) I , } D = 4 - G • ( 1 + l ~ ) 2 (1+/2)8 1 + 1 ~ ;

R 2 - - • - - l • (1 + l 2 - - 2 t l f Q )

K = u _ l i r~ . . !_ l 2 ( l _ l V ~ + 1 2 ) 2 - -

RS • 2 1 5 x ( l - - I 2)

L = ~ [ , i - - T F ~ - ~ t ' - - ( l _ t V ~ + t ~ y

• 1 3 l - - 2 l - - l - - - ~ .

~2. .,p2 - ( ; - - ,F?T-z~ J'

12

1/2) / ~ • Rz z I - - l "--T ' F = l +

M = 4-G (l - - l t / 2 @/2) ' 4 - G ' ~ g "

Solving (20) and (21) s imul taneous ly , we get

cos 20 (U1C - - U ~ B ) _L sin 20 .UI .D ~I,z -t- (I1 ~- cos 2 0 (AC - - BF) -!- sin 2 0 . A . D '

(23)

whence we get

AU2 - - FU1 ~ 2 - - ( I 1 = c o s 2 0 ( A C - B F ) @ s i n 2 0 . A . D ' (24)

1 (U1C - - U2B) cos 20 + U1D sin 20 - - AU.,4- FU1 01 = -if" (AC - - BF) cos 20 ~- AD sin 20 '

1 (U1C - - U.zB) cos 20 1- U1D sin 20 + AU.a-- FU1 (Ia = -~" (AC - - BF) cos 20 -[- AD sin 20

To s impl i fy the computa t ions , i t wil l be c o n v e n i e n t to write the expressions for o t and o z in the form

I U 1 - - U2 (A + B cos 20) al = y " ( A C - - BF) cos 20 + AD sin 20 '

(25)

(26)

(27)

Ux - cq (A - - B cos 20) (28) O'2 = A + B cos 20

We find the ang le O d e t e r m i n i n g t h e d i r e c t i o n of ac t ion of o l and o9. in the plane of the measurements from

(22), (23), and (24): 1 a

0 = ~ a r c t g -if, (29)

where

a = [ ( A C - - - B F ) U 3 + ( U , F - - U 2 A ) L - [ . ( U 2 B - - U I C ) M ] ,

b =- [ U I D M + ( U 2 A - - U 1 F ) K - - U a A D ] .

Thus the required values are ca lcu la ted as follows from measurements , made with a m u l t i c o m p o n e n t defor - meter , of the radia l d i sp lacements of the contour of a smal l borehole, due to load re laxa t ion of the rock near i t by a l a r g e - d i a m e t e r borehole , 1) De te rmine (9 from Eq, (29); 2) find the stress componen t o t from Eq. (27); and

f inal ly , 3) c a l cu l a t e a 2 by means of Eq, (28),

In a par t icular case, a system of two equat ions of the form of (19) enables us to d e t e r m i n e the values of the pr incipal stresses in e lements of the rock mass, provided that we know their d i rect ions of ac t ion , For this purpose we must measure the rad ia l d i sp lacements of the contour of the s m a l l - d i a m e t e r borehole in the d i r ec t ion of ac t ion

of the pr inc ipa l stresses, i .e , , with O r = 0 and Oz = ~/2. In this case,

U1 (A2 + B2) - - U~ (At + Bt).; (30) - - A2B1 - - AzB,,.

AzU 1 - - AxU 2 ~ 2 - - A 2 B 1 - - A x B 2 ' 0 1 )

where

2 (1 - - 12) ] 6 $-z-~ j;

RZ~ [1 + ~ f 1 1 ) R~ 1 A1 = Td- [ - - i f - ~,1 - z ~ (1 - - 0 3 ;

B1 = R2• 1 2---G" " 1 - - l ;

R2• [1 -[- ~ { - -X 1 ) R~ (312 - - 1) A2 = ? - 0 - [ - ' g - ( f ~ - - - l ~ V ( l + l * ) ~

B2 R2n I - - 2----~ �9 1 _~_/2.

i a

In a l l the formulas the quantities R, l , U t, U 2, and U 3 are dimensionless (normalized to the radius r of the smal l borehole) ,

The range of appl ica t ion of the above method of de te rmining the stresses in the rock mass depends on the parameters R. l , E, and v , If the modulus of e las t ic i ty and Poisson's rat io of the rock mass are fixed parameters , not depending on the experimenter, then the values of R and l can be varied with cer ta in ltnits,

From an analysis of the theory [3] on which the method and Eqs, (25) and (26) are based, i t follows that the most favorable conditions for de termining the stresses will occur when the dis tance between the boreholes (s = 1 ~ R - r ) is greater than four times the radius of the smal l hole and less than the radius of the large hole, Outside these l imi t s we find cases in which the accuracy of the stress de te rmina t ion method is lower,

We must also st ipulate the dimensions of the boreholes, becausse for cer ta in va luesof R/r tensi le stresses will arise near the smal l hole, and the resistance of rock to these is poor, The op t imum value of R/r when s = (4-5)r is given by R/r ---10, For the commonest ease, in which the d i ame te r of the measurement hole is 46 mm, the r e l axa - tion hole should be between 184and 460 mm in d iamete r ,

Determinat ion of the stresses arising in the rock mass has an error due to random errors in the measurements

of U i, P,, I, E, and v, The greatest re la t ive errors are due to the modulus of e las t ic i ty (10-15%), Poisson's rat io (1-3~ and the displacements (up to 2~ The total error of the method is 15-20~

L I T E R A T U R E C I T E D

1. M, V, Kurlenya and A, V, Leont 'ev. "A method of de termining the stresses in sedimentary rock in si tu." Author 's Cert i f icate No, 368.402; Byull, Otl~rytiya. Izobreteniya. Promyshlennye Obraztsy. Tovarnye Znaki , No, 9 (1973),

2, N, I, Mnskhelishvili. Some Fundamental Problems in the Mathemat i ca l Theory of Elasticity [in Russian], Nauka. Moscow (1966),

3. A, S, Kosmodamianskii . in: Tr, VNIMI. 422, Leningrad (1961),

14 �84