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重复压裂SPE


J

Sp(.Y-”

A Study of Inclined
ABBAS AL I DANE SHY MEMBER SF E.AIME

Hydraulic

Fractures
SERV/CES

I

HALLIBURTON


DUNCAN, OK/-A.

ABSTRACT
The results of o theoretical and experimental investigation 0/ inclined hydraulic /rac[ures, reported in this pcper, indicate that such fractures d~ not genera [[y initiate perpendicular to the rruximum tensile stress induced on the borehole wall. Unlike axial or normal hydraulic fractures, a degree of shear failure seems to be ossoc iated u,ith tbe initiation an(i extension 0/’ almost all inclzqed )Jydrauf ic [ractures. These fractures o~ten intersect tlIc borehole along t~(o diamet~icrdly opposite oxial fines, thus .git)ing i: the crppcarance 0/ an axial fracture. !nc[ined Lydraulic fr~ctures generally ci]ange the!r orientation as they e::tcnd away from tl]e tce~lbor(’ until they become perperrdicu[ar ta the lczst compressive in-si~u principa? stress. Ti]ercjore, the b,. ”ehole trace of such frrzctrmes cannot be u~ed for their positive identification,

INTRODUCTION The process of hydraulic fracturing of a formation essentially consists of injecting a fluid inside the aorehole and pressurizing it urtil the induced stresses exceed the strength of the formation and cause failure. Failure is generally indicated by a sudden major drop in the variations of the borehole fluid pressure with time. In general, in an isotropic medium, the over-all plane of a hydraulic fracture is either parallel, inclined, or perpendicular to the axis of the borehole from which it is extending. Accordingly, these fractures will be called axial, inclined or normal, respectively. This classification of hydraulic fractures refers them to the borehole where they are observca rather than the ground surface or the bedding planes. In case of vertical boreholes, axial and normal fractures become identical with vertical and horizontal fractures (which are the terms often used in petroleum industry). In the first comprehensive analysis of the mechanics of hydraulic fracturing, Hubbert and Willisl proposed that axial or normal hydraulic fractures initiate when the maximum tensile stress induced around the borehole exceeds the tensile
Paper (SPE 4062) was prepared for presentation at the Sp EAIME 47th Annual Fall Meeting, held in San Antonio, Tex., Oct. 1973 American Institute of Mining, 8-11, 1972. @_) Copyright Metallurgical, and Petroleum Engineers, Inc. preferences given at end of paPer. APRIL, 1973

strength of the formation, and that such fractures extend in a plane perpendicular to the lecst stress. compressive in - situ princip31 The correctness of this proposal has since been verified and Fa]rhu:st,2 who conducted an by Haimson extensive series of laboratory experiments on the subject. In their theoretical and experimental work, Haimson and Fairhurst assumed that one of the in-situ principal stresses is parallel to the borehole axis. Under such a condition, one can only create an axial or a normal hydraulic fracture in an isotropic medium. For the case when none of the in-situ principal stresses are parallel to the borehole, Fairhurst3 derived mathematical expressions for the stress components on the borehole wall, in isotropic and transversely isotropic media. Experimentally, von Schonfeldt4 ~ijd Daneshy5 independc!}tly observed that under such a condition the fracture orientation is influenced by the borehole in its vicinity. The trace of inclined hydraulic fractures at the wellbore was found to be misle~ding if used for the purpose of determining the eve;-all fracture orientation. The research reported here is an extension of a previous work on the subject of inclined hydraulic fractures. It includes the computation of the magnitude and the orientation of the maximum tensile strsss induced at the borehole wall, for each experiment, and the resulting fracrwe shape. Such investigations can, in the course of time, provide means of determining the over-all fracture type at :reat depth, which has significant importance in many fields, such as geophysics, petroleum, geological and civil engineering. STRESS DISTRIBUTION AT THE WALL OF THE BOREHOLE total Let 01, 02 and 03 be the three in-situ principal stresses whose values and directions are assumed to remain constant throughout the isotropic porous elastic formation under consideration. No restriction is imposed upon the direction of any of the in-situ principal stresses, except the mathematical requirement that they should be mutually perpendicular. Consider a coordinate system 0X1X2X3 chosen such that 0X3 is the borehole axis, and Oxl lies in the plane 0U102, Fig, I. The orientation of Oxl X2X3 with respect to 001 U2U3 can be expressed in terms of its direction cosines, Cii, which are

trigonometric functions of two angles, defined as the following (Fig, I): ~ = the angle between 0X3 and 003 /3 = the angle between the projection 00102 and Ool axis

@ and

~,

TABLE

1 -

DIRECTION

COSINES

OF .,,.2,3

WITH

RESPECT

TO CU, fW3

of 0X3 on
in which A = ~cos2$~ t sinz$ COS2,6

The mathematical expressions for Cij are given in Table 1, The components of the total stress tensor with respect to OX1X2X3, namely, Sij, are related to the in-situ principal stresses by the expression

u= rr

-PW’

...””””””(3)(3)

s~j =

z
‘ij

Cik Cjk ok
stresses,

...
Oij ,

’00
.(1)

=

’11

+

’22

-

2

%

-

C22)

q

cos2e
=0

- 4(J12sln20
-

+ Pw - a,, )

.

.(4)

The corresponding effective (compression negative): +p~6~J10.. ‘ij . =

are (J

. .(2)

Zz

33

2V ‘(0,1
1 +

where p. is the reservoir fluid pressure (pressures are considered positive) and ~ij is the Kronecker delta. The presence of a cylindrical borehole and the application of fluid pressure inside it will change the stresses that used to exist at the vici,:ity of the borehole prior to its drilling. With reference to a cylindrical coordinate system, rflz (borehole axis being z-axis), the state of stress is defined by the three normal stresses or, , o~j~, UZZ and che three shear stresses 0r~,~J(]z,~r2, Fig.2. The ma:hematica] equations for these stresses are given in the Appendix for the case of an impermeable formation or an extremely viscous fluid (zero fluid penetration into the wall). The corresponding expressions for the stresses at the wall of the borehole are derived from Eqs. A-1 through A-6 by substituting ? = (~ in them. This will yield CT.

q

COS2fl

201Z sin20

1

. . .(5)

‘r9

= 0..............(6) cosf3 - u13 sin O)

%3z = 2 (a23 u= rz
From Eqs.

. . . . . . . . . . . . . . . . . . (7)

0.............(8)

U* ,,
,’

3 through 8 it can be seen that, since = O, therefore, rJr, is one of the secondary ‘r{] ‘ ‘r.z prilicipal stresses at any point on the borehole wall (Fig. 2). Since pa, is a positive entity, therefore, rTrr is a compressive stress (negative sign in ~lq. 3). The other two secondary principal stresses lie in a wall and can be plane rangent to the borehole calculated from Eqs. 4, 5 and 7. In a permeable formation the effective stress components around the borehole are calculated by adding the stresses induced due to fluid penetration to the corresponding stresses given in Eqs. A-1 through A-6. For the case when the :reatment fluid has the same viscosity as the reservoir fluid, Haimson and Fairhurst2 have solved the problem in terms of integrals of a function p (T,t) denoring the variations of fluid pressure with dis:ance r and time t. At the wall cf the borehole these expressions have simple forms and, when added to Eqs. 3 x, I X2
Cree *
M ,

x,
r

,Vrf
U*Z O&,

e

0’

x,

CT”
FIG, 1 — 0F3ENTATION

I X9
FIG. 2 — STRESS COMPONENTS WALL.
PET ROLE1” M

@
ON THE BOREHOLE

OF OX, X2 X3 WITH RESPECT TO THE IN-SITU PRINCIPAL STRESSES.

62

sOCIETY

OF

EXGIXEEBS

JOURNAL

through

8, yield:

from the expression

urr =0.
%8

. “””””””’’”””(9)

‘%1
q

‘la
2 -a 1
1

-2

%

-Q
+
“ PO)

c0s2e - 4U12 sin2fl

(Ueo - (7=2)2+ 4UOZ2 .
i . . . . . . . . . . . . . . . . (15) The angle yp between is equal to up and the borehole directrix

(

- 2V -v )

(Pw

. . . . . . . . . . . . . . . .

(lo)
YP=2
As

~ tan-l

-

2%Z

.

.

(16)

%0 - ‘Zz

COS28

+

2U12 ~in20

1
. .

+

(
.

l-al-2v
1 -v
. . . . . ..’

)
. .

(Pw-

Po)

.

.

.

(11)

‘r(l=
‘ez =

0 .

. . . . . . . . . . ..

(12) .

2 (U23 sinf3) . . . . . . . . . . . . . . . . . (13)
COSO U~,

c1

rz

=0

.

. . . . . . . . . . ..

(14)

Where a is a constant of the potous elastic rock related to rock matrix and rock bulk compressibilities C, and Cb by the expression

Point A! travels on the circular periphery of the borehole, the values of 000, Uzz, ~oz and, consequently, Op and yp change. The maximum tensile stress on the borehole wall, Um, is equal to the highest value of up for all points. tMathematically, setting the derivative of up with respect to d equal to i:ero will yield the values of O - On corresponding to cxtrerne values of rr . Substitution of Om in Eqs. 15 and 16 will yiel i Om and the corresponding arlgle ym. From Eqs. 3 through 8, or 9 through 14, it can be seen that every two points identified by O and O I n (diametrically opposite) will have the same UOO, Uzm and the same Ooz, but with opposite signs. Eq. 15 shows that the sign of Ooz doe:: not play a role in the value of up since Udz appears in a quadratic form. This means that if Om occurs at (l = Om it will aISO occur at O - Om * 7:. The loci of points with O = cm and O = On, * n around the borehole are two diametrically opposite straight lines that are the loci of maximum tensile stresses. The corresponding angle ym, however, does not have the same value on these two lines. J3ecause of the difference in the sign of cr~=, utilizing Fq. 16, it can be shown that

Eqs. 9 through 14 show that the only non-zero secondary effective principal stresses at any point on the borehole wall lie in a plane tangent to the borehole at that point.
MAXIMUM

=-

Ym

e
. . . . .
. .

=Bm+ll

.

.

.

.

.(17)

TENSILE STRESSON THE BOREHOLE WALL

EXPERIMENTAL

METHOD

As discussed earlier, it is generally believed that initiation of hydraulic fractures occurs when the maximur,l tensile stress induced on the borehole wall exceeds the tensile strength of the formation. Let M be a point on the borehole wall under the influence of normal and shear stres’ses given by Eqs. 3 through 8, or9througfi 14. of the three shear stresses at this point only trdz has a nonzero value. Since one of the three principal stresses at Point M, namely, Urr, is always compressive or zero, therefore, the maximum tensile stress at this point, Op, will have to lie in a plane tangent to the borehole at h!. Mathematically, up can be calculated Aprrr r.,
1973

Three rock types were used for the experimental investigation of inclined fractures. These were Carthage limestone, limestone Indiana and Hydrostone (35 parts water/100 parts Hydrostone by weight). All specimen were 6 x 6 x 10 in. To simulate the in-situ principal stresses, each was pressurized perpendicular to its specimen faces before it was hydraulically fractured. The result was three independent mutually perpendicular pressures, Fig. 3. Since the orientation of these pressures was fixed with respect to the specimen, to obtain inclined fractures the borehole had to be
63

~6

in. ——

I I

I
1

I I

Open .--*

Hole -h , II’ I j,) It’ / ,, : I 1 , t
I //’ --

,

,;! /’/’ / 1;1’ 1! ‘/
,’!

-----

----

---——

$“””

.+

P-’”””’

~

~

/“’”

OF

3 — SAMPLE GEOMETRY AND THE DIRECTION HYDRAULIC EXTERNAL PRESSURES FOR FRACTURING EXPERIMENTS.

drilled at an angle (Fig. 3). This was done by the aid of a specially built drilling frame. The function of this frame was to guide the drill in such a way that the resulting borehole would be inclined at prescribed angles r/I and /3 with respect to the direction of the external pressures and would also through the center of the specimen. All pass boreholes were drilled throughout the height of the specimen and were 0.313 in. in diameter. Two steel tubes, each ? in. long, 0.313 O.D. and 0.235 I. D., were cemented at the top and the bottom of the borehole (Fig. 3). This borehole arrangement yields reasonably uniform stresses throughout the open-hole section and eliminates unpredictable stress concentrations which would have existed if the bottom of the boreholes had terminated inside the specimen. To study the influence of various borehole

inclinations (with respect to the in-situ principal stresses) on the type and the borehole tra~e of ~he resulting hydraulic fractures, the boreholes were drilled at different angles @ and ~. These were ~ = 0°, 1~“, 300, 45°, 600, 75°, for each of which three samples were prepared with t$ = 8°, 16° and 24°. There was one additional sample prepared with /3= 45°; q5 = 32°. The limited number of variations in ~ was mainIy due to the size of the specimen that did not allow any larger 4’s. The total number of combinations tested with each rock type was 19. Hydraulic fracturing of the samples was achieved by injecting fluid inside the boreholes at a constant rate. For Carthage limestone and Hydrostone, the injection rate was 2.27 cc/see. For Indiana limestone this rate was not sufficient and had to be changed to 17 to 23 cc/see before any fracturing could be obtained. FGr all experiments, the fluid was injected by an MTS close-loop press ure-strokcregulator. With this instrument one can hydraulically fracture a spscimen in the laboratory such that or injection rate will follow a either pressure presc:ihed function during the test. Both the pressure and the total volume injected were monitored during each experiment and their variations were recorded against time on an X. Y recorder. The permeabilities tc nitrogen of the three rock types used were 0.004 md for Carthage limestone, 15.12 md for Hydrostone and 146.52 md for Indiana Although nitrogen was not used for limestone. fracturing the samples, its use for permeability measurement yields more accurate results and allows a comparison between the three rock types. The porosities of these rocks were 18 percent for Indiana limestone, 28.06 percent for Hydrostone 35/100, and 1.14 percent for Carthage limestone. Each of the numbers given above is the average of five measurements on five different samples. EXPERIMENTAL RESULTS

Table 2 shows the results of some of the experiments carried in Hydrostone, Carthage limestone and Indiana limestone samples. The

TABLE E;p;;~~nt I F-3 IF-12 IF-10 Sample Type HS’ HS HS HS

2 — SOME OF THE # P k!w!?S2k!w@_w_. 24 8 24 24

RESULTS ml

OF INCLINED [J* (psi) —(~ji ) -500

HYDRAULIC & 5, 32C

FRACTURING om
Y;

EXPERIMENTS
‘Jm

(degree)

(degree)

(psi)

Fracture axial oxi~l oxiol axial axial a:ial
axial

Trace (A) (E) (C) (B) (B) (B) (B) (C) (@) (E)

0
15 45 15

-500
-2,200 -1,200 -800

-1,200
-2,500

- l,OCO 2,325 -400 -500 -1,300
- 31XI

90.0
92.8

-8,5
14.2

2,773,1
3,578.2

and inclined and inclined
ond inclined

-2,000
-1,200 -1,200 -1,800

IF-14 IF.22 IF-23
IF-33

CL S** CLS
CLS

24 8
8

0 15
60

-300 -1,500
-1,500 -1,500 -200 -200 -800 -800 - Soo

3,060 3,200 4,730
6,900

99,3 91.9 90,0
91.4

15.9 5.0 -8.0
4,2

2,956.2 3,448.4 4,638.1
4,287.2

oxiol

(A)

IF.34 IF-39
IF-40 IF-43 I F-44 I F-57 The letters

CL S
lLSt ILS l~s ILS ILS

16
o o 15 15 75

60
8 16 16 24 24

-2,000 -200
-400 -400 -1,300 -1,300 - i,300

-200 -200
-600 -600 -200 -200 -200

5,690 6,100
— – — — —

92.1 99.1
-

3.6 7.4
-

3,196.1 3,651.6
— — — — — 4.

and inclined rodial (F) and inclined axial (A) and inclined and inclined and inclined and inclined

oxial axial axial

A, B, C, D, E and F mean that the fracture

trac ~ is sinsilcr tO the corresponding

part of Fig.

q HS = Hydrostone 35/100. q*cLS = Ca,thage I imestone.
flLS
6%

= Indiana

limestone.
SOCIETY OF PET ROLE[l M ENc IS EERS JO URXAL

parameter PC in this table denotes the breakdown pressure. The anqle y; is the complement of the angle ym, i.e.,

Y; =-.

w 2

Y~”””””’””””

(l@

Physically, this is the angle between the normal to the borel,ole directrix and the crm direction. As discussed earlier, y; has opposite signs at O = (?m and d = @m + rr, and, therefore, its importance will only be in its absolute value. The values of f?m, y; and am displayed in the tables were computed from Eqs. 3 through 8, or 9 through 14, along with Eqs. 15, 16 and 18. The choice depends on whether the material is considered impertreable to the treatment fluid or not. Under the heading “Fracture Trace’ the traces of the created hydraulic fractures at the wellbore are described. Here, the capital letters in parentheses refer to the corresponding parts of Fig. 4 and mean a similarity between rhe two traces. For example, the trace of the hydraulic fracture induced in Experiment IF-12 (Table 2) is described as axidl and inclined (E). This means that at the wellbore the fracture trace had axial and irrclined components similar to Part E of Fig, 4. The maximum tensile stress, on, and the in Table 2 are corresponding dm and y; presented those computed from theoretical considerations for corresponding values of p=. The material properties

A

B

c

.,>, ,, ,.

,! ::
?
1

Fracture Trace

}.J ...
3

D

E

F

R P
l\
~ :,
., . Frocture Trace
I

U
1“-’

if
FIG. 4—VARIOUS

TRACES

,. ..., d

::

.. ...

needed for these computations are Poisson’s ratio, v, and the Biot’s constant, u. The values of v were measured experimentally and were 0.215 for Hydrostone and 0.33 for Carthage limestone. The value of a = 0.82 for Hydrostone was taken from Haimson.G The theoretical values of y; were never found in agreement with the experimental values, except for y; = O. Almost all experiments yielded a trace with an axial component (except for two in Carthage limestone) which correspond to an apparent zeio value for y;. In cases when fracture traces had inclined components, the angle of inclination of the fracture trace was riot constant sad, therefore, could not be compared with theoretical values. A simi Iar situp tion exists regarding theoretical and experimental values 0! Om. In most experiments @m varied at the vicinity of 90° and 270°, but its acrual value could not be measured with reasonable acccracy due to the roughness of the fracture traces. However, since most of the computed values of On were also not appreciably different than 90° and 270°, It may be concluded “ that the rheorerica} and experimental Om’s were in agreemenr. No definite patterns were observed for the variations of on with changes in either qi or /3. In Hydrostone samples the average value of am for all tests for which UI > 02 and cr3 was 2,789 psi and for us > U2 and UI it was 3,?:0 psi. Cm the other hand, in Carthage limestone the trend was the opposite. In this rock Um (average) was equal to 4,564 psi for al >02 and 03 and 3,63] psi for 03 ,> rT2 and O1. The experimental investigations conducred on Indiana limestone samples were limited to a simple observation of fracture traces and types corresponding to various combinations of the external pressures The reason was thar the applled to the specimen, paramerer a is not known for Indiana limestone. hydraulic Regarding the orientation of the fractures, it was observed that in most tests rhe initial fracture induced was an axial one which subsequently changed its orientation favoring a propagation perpendicular to the least external pressure applied to the specimen. It is believed that, had the samples been larger, all hydraulic fractures would have become totally perpendicular to the least externally applied pressures. to investigate the influence of the Finally, borehole pressurization rate on the breakdown pressure, PC, four tests were run on each of the three rock types used in these experiments. As shown on Fig. 5, the breakdown pressures generally increased with an increase in pressurization rate. This result is similar to what HaimsorrG had obtained for purely axial and normal tractures. Fig. 6 shows some of the hydraulic fractures created in these experiments. GENERAL DISCUSSIONS AND CONCLUSIONS

[

OF INCLINED

FRACTURES

AT THE WEJ.LBORE.
APRIL. 197.?

The sequence of events leading to the creation of inclined fractures is usually more complicated
6S

than for axial fractures. The source of difference is the existence of shear stresses on the borehole wall that make the maximum tensile stresses different than axiai or tangential stresses. The results of most theoretical and experimental works on hydraulic fracturing indicate that both initiation and extension of axial fractures are caused by purely tensile failures of the borehole wall and the surrounding formation. As a result, such fractures possess relatively smooth faces and borehole traces. Any roughness on the traces of these fractures is caused by local inhomogeneities. Fractufe faces exhibit only those facial markings particular to purely tensile failures. The hydraulic fracture propagates in only one plane, with no difference in its orientation close to or away from the borehole. The initiation of inclined fractures is somewhat different than axial fractures. The borehole traces of inclined fractures are usually rough and very seldom perpendicular t~ the maximum tensile stresses induced on :he borehole wall. This observation plus the existence of shear stresses around the borehole (as shown by Eqs. A-1 through A-6) suggest that inclined hydraulic fractures are initiated as a result of shear as well as tensile Further proof of the failure of the borehole. occurrence of a shear failure is the existence of steps* on the faces of inclined hydraulic fractures as shown in Fig. 7. This photograph shows the face of an inclined hydraulic fracture along with its facial markings. Every inclined hydraulic fracture undergoes a certain amount of reorientation once it extends from sufficiently away the borehole. This reorientat; .rn is necessary for the fracture to become compressive in-situ perpendicular to the least — *steps are small, ste.sply inclined offsets of the main fracture
and denote a shear mode of failure at those locations where they occur. The layout of steps is usually used as a guide toward determining the path of fracture propagation.7

ING EXPERIMENTS,

7000

.
~ c no J ~ m @ li g ~
0 e ;

6000

5000

4000

3000

2000 200 500

1000

2000
Rote,

5000
psi/ wc

10,000

Borehole FIG. 5 —

Fluid Preswrizotlon

SURES

VARIATIONS OF THE BREAKDOWN PRESWITH BOREHOLE FLUID PRESSURIZATION RATE.

FIG,

7



FACIAL FEATURES OF HYDRAULIC FRACTURE.
OF PEIROLSI’M

AN INCI.INED

66

SOCIETY

EXCIXEER5

JO I’RSAL

princi~al stress. For most industrial and scientific . . applications of hydraulic fracturing a knowledge of the orientation of the over-all plane of fracture is very useful and at times even essential (e. g., in-situ stress measurement by hydraulic fracturing). But unfortunately there is still no economic method available to determine the location of a hydraulic fracture at points away from the wellbore. Instruments such as impression packers or borehole televiewers only recoid the borehole trace of a fracture and therefore give no definite information : bout the fracture location away from the borehole. Much research is needed before important questions concerning inclined hydraulic fractures can be answered satisfactorily. In particular, methods of recognizing these fractures at great depths are most needed for many branches of engineering. At the present ~ime there is no method available to iso!ate the inclined fractures from the many axial traces observed in fjeld experiments. Although there is no doubt that part of the industrial hydraulic fractures are inclined, the relative importance of the subject depends largely on the frequency of their occurrence, a factor yet unknown. NOMENCLATURE borehole radius

C5ij

=

Kronecker

delta Oxl axis and the radial a point on the borehole to am rock stress effective to 0X1X2X3

e=
flm J). =

angle between line through wall value Poisson’s

of O corresponding

ratio of the formation

‘ii

=

of the components tensor with respect

um = maximum tensile stress on the borehole wall maximum tensile stress at a point on ‘P ‘ the borehole waII
(7,,,000, tangential :- radial, on components 0 ZZ

and axial the borehole

stress wall,

respectively
0,0 ,q?= > = shear 0 ‘L wall angle

stress between

components

on the borehole

0X3 and or73

ACKNOWLEDGMENTS The author would like to thank the management of Halliburton Services for permission to publish Pittman, this paper. Thanks are also due Forrest Tony Giroux and James Mueller for their assistance in the design and manufacturing of the equipment, and David Meadows for helping to conduct the experiments.
REFERENCES

rock bulk compressibility direction cosines of ox1x2x3 with respect
tO 0U~0203

rock matrix

compressibility wall Hydraulic Fracturing, ,, 7.,ansr, AIME (1957) VO1, zl”~ 153-168, 2. Haimson, Bezalel and Fairhurst, Charles: “Hydra,~lic
Fracturing in Porous-Permeable Tech. (July, 1969) 811-817. Materials, ” -/. Pef. 1. Hubbert, M. King and Willis, David G.: “Mechanics of

point on the borehole

coordit]ate system reservoir fluid pressure function denoting the variations of fluid pressure with distance from borehole axis and time borehole f{uid pressure distance between a point formation and the borehole inside axis the tensor

components of the total stress with respect to OX1 X2.Y3 time

C.: “Methods of Determining In-Situ Rock 3. Fairhurst, Stresses at Great Depths, }> TR 1.68, Missouri River Div., Corps of Engineers (Feb., 1968). Study of OpenH.: “An Experimental 4. von Schonfeldt, Hole Hydraulic Fracturing as a Stress Measurement Method with Particular Emphasis on Field Tests, ” PhD thesis, U. of Minnesota, Minneapolis (Nov., 1970). of A. A.: “True and Apparent Direction 5. Daneshy, at Hydraulic Fractures, ” paper SPE 3226 presented the Fifth Conference on Drilling and Rock Mechanics, Austin, Tex. (Jan. 5-6, 1970). Fracturing in porous and B.: ltHYdraulic 6. Haimson, Nonporous Rock and Its Potential for Determining lnSitu Stresses, ” PhD thesis, U. of Minneso!. a, Minneapolis (July, 1968). Fracture Mechanics from 7. Lutton, R. J.: “Tensile Rock Surface Morphology,”” Dynamic Fracture Mechanics — Twelfth Symposium on Rock Mechanics, U. of Missouri, Rolls (Nov. 16-18, 1970) 561-571. and Triaxial 8, Leeman, E. R.: “The CSIR ‘Doorstop-er’ Rock Stress Measuring Instruments, ” Rock ~~fecha>~ic.% (1971) Vol. 3, 25-50.

three in-situ principal stresses Biot’s constant of a porous elastic formation angle between the projection of ox3 on Oul cr2 and cwrl axis between angle directrix complement between angle directrix cm and the yn, the borehole borehole

of the angle up and

APPENDIX STATE The stresses
APRIL, 197.?

OF STRESS a pressurized

AROUND A PRESSURIZED borehole are induced

BOREHOLE by the three in-situ principal stresses
67

existing

around

and the axis are viscous ratio, u,

fluid pressure, pw, inside the borehole. .4ssuming that the displacements parallel to the borehole zero and that the fracturing fluid does not penetrate the formation (impermeable formation or very fracturing fluid), the stress components around a borehole drilled in a formation with Poisson’s are given by + 022 l-~+ ul’ -022

u=

rr

all

l+3d
(

2

(

r2

r’

.4& r2 )

cos2e

+ U12 (

l+3f’h
=a2
r2 all

%0 =

r
12

i’”+ -[%:1 “-’)
~22 ~ () +

011

+

U22

()
l+—-

2

3 —

ah

rb

cos2e

+1+3$) sin20]+[Pwq... . . . . ..(’-2) . .
u=

Zz

[{’33

-v

2 (u,~ -62*)

a2 — cos2e r2

a2 + 4012 — sln28 r’

(A-3)

}]

In each of Eqs. A-1 through A-6, the terms inside the first brackets are those induced by uij (Leeman8) and those inside the second brackets (whenever present) are the stresses induced by the borehole fluid pressure. The term r is the radial distance between the point under consideration and the origin of the cylindrical coordinate system n%-, which is the center of the circular cross-sectioa of the borehole. ***

“re=r’’i”22 (1-’$+2$)sin+”,2t2 $+ ’f) ’f) 1 ( ( )1
COS2(3............ ...................
. . .. (’-4)

a2

‘6Z=

‘-a~3s~ng+U23c0so)

1+=

““”’””””””””(A-5)

[

)1

‘rz =

(a,3’ cos8+u2$sin

O)

l-—

a2 r2

. . . . . . . . . . ..

(A-6)

[

6a

SOCIETY

OF

PET

ROLEl~M

ENGINEERS

JOURNAL


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