Well logging
Content
APPLICATION OF ROCK MECHANICS IN HYDRAULIC
FRACTURING THEORIES
Abstract
Three aspects of rock mechanics have been of great
value in developing hydraulic fracturing theories, viz. :
(1) Elastic behavior. (2) Shear failure criteria. (3) Concepts of fracture propagation.
Within the framework of these concepts, the following topics will be discussed: (i) The relationship
between rock matrix stress and total stress. (2) Estimates and significance of in situ earth stresses. (3) Factors influencing fracture orientation. (4) Fracture
extension pressures. ( 5 ) Apparent rock surface energies. (6) Fracture widths.
Résumé
Trois aspects de la mécanique de la roche ont largement contribué au développement des théories de la
fracturation causée par une force hydraulique. . .
(poussées d’eau) . c.à.d.: (i) Le comportement
élastique. (2) Les critères des efforts de cisaillement qui
causent la faille. (3) Les concepts de l’extension de la
fracture.
C‘est dans le cadre de ces concepts que les sujets qui
suivent vont être discutés: (1) Les rapports entre les
efforts de la matrice de la roche et la somme des efforts
(2) Evaluation et signification des efforts terrestres sur
les lieux (“in situ”). (3) Les facteurs qui influencent
l’orientation de la fracture. (4) Les pressions qui
résultent de l’extension de la fracture. (5) Les energies
évidentes à la surface de la roche. (6) Les largeurs de la
fracture.
INTRODUCTION
departure from idealized linear behavior, and for many
problems the linear elastic theory must be augmented
with additional concepts or approximations.
.
One of the significant advances in petroleum technology within the past twenty years has been the
development of hydraulic fracturing processes. Within
this same period, understanding of the mechanics of
Limits to idealized behavior
rock behavior has increased. It is the purpose of this
paper to show how knowledge of rock mechanics has
There is clearly a limit to elastic behavior if a
been applied in the understanding and further develop- sufficiently large uniaxial tensile stress is applied (the
ment of hydraulic fracturing processes.
tensile strength). Also, if a combination of stresses is
Those aspects of rock behavior that will be of interest applied, there is a limit beyond which shear failure will
are briefly summarized, and then in situ earth stresses occur. For present purposes, it is sufficient to describe
and factors influencing hydraulic fracture initiation this envelope of shear failure by the well known Mohr
and extension are discussed.
failure theory.’ If stresses are not too large, the shear
limit can be written in a simple manner as shown by
equation (i).
BEHAVIOR OF ROCKS
r = C‘+un tan q5
(1)
= shear stress at failure
Many investigators have treated rocks as linear,
C = unit cohesive strength
elastic materials. Such an assumption permits the
powerful mathematics of the linear elastic theory to be
un = stress normal to the plane of failure
applied in solving problems. For many circumstances,
4 = angle of internal friction.
this approach appears to be entirely justified, and
meaningful solutions can be obtained. On the other Within the region of shear stability, the rock behavior
hand, studies of rock behavior generally show a is presumably elastic. However, studies show at least
four additional complexities :
(i) For rocks of low to moderate strength, there is
a pronounced hysteresis for moderate stress changes.
by T. K. PERKINS, Atlantic Richfield Co., U.S.A.
15
Rock Mechanics in Oilfield Geology Drilling and Production
76
12
~~
-- -z
<n
REDUCTION
IN PORE SPACE, PER C E N T
Fig. 1-Compressibility behavior of Bartlesville
sandstone (from Carpenter and Spencer2)
Fig. 1 illustrates this type of behavior with data of
Carpenter and Spencer2 showing the compressibility
of Bartlesville sandstone. The hysteresis obviously
indicates other than purely elastic behavior.
(2) For large stress changes, elasticity of granular
materials is at best non-linear. See Deresiewicz3 for a
review of mechanical behavior of granular media. As
an illustration, Brandt4 has computed the volumepressure relationship for an aggregate by applying the
Hertz theory for the deformation of elastic spheres.
Fig. 2 shows experimental data together with Brandt’s
computed curves.
1.0
0.90
0.96
k
0
8-
fn
m y w>
o
W n .a
œ
g3 Eo
-
4-
bfn
<n
.~
a -0
+
z
w
o
u)
z
w -4
(4) When large stresses are first applied to unconsolidated materials, there is generally a distinct
“compaction” behavior which is different from subsequent “elastic” behavior. Fig. 4 shows the data of
Botset and Reed6 which illustrates an initial severe
compaction of unconsolidated sand. Repeated cycles
0.94
POROSITY A T PRESSURE
POROSITY A T ZERO PRESSURE
Fig. 2-Porosity
variation as a function of pressure
(from Brandt4)
Curve A-Basal Tuscaloosa, Miss. ;Porosity 15 %
Curve B-Basal Tuscaloosa, Miss. ;Porosity 24 %
Curves C & D-Theoretical Curve
Curve E-Southern California Coast; Porosity 25 %
Curve F-Southern California Coast; Porosity 22
35
36
37
38
39
40
41
42
POROSITY. PER C E N T
Fig. 4-Compaction of unconsolidated sand
(from Botset and Reed6)
II
Rock Mechanics in Oilfield Geology Drilling and Prcduction
of loading show continued crushing as well as “elastic”
behavi or.
Even though rock behavior is complex, for certain
stress ranges and for certain kinds of problems, application of the linear elastic theory will given meaningful
engineering solutions.
metry, however, it seems likely that these various
effects would be limited to a region close by the wellbore, and that generally in the reservoir the weight of
the full overburden would bear.
Effect of interstitial fluid pressure
There are two broad schools of thought concerning
the prediction of in situ horizontal earth stresses. In
effect, with one approach it is supposed that the
environment and state of earth stress are known from
the time of initial mineral deposition. In using this
approach,l7J0.21 it is commonly assumed that behavior is purely elastic, that elastic constants were not a
function of stress (depth) or age (degree of cementation) and that there were no tectonic influences (a condition of zero horizontal strain is generally assumed).
If grain compressibility is small compared to bulk
compressibility, these assumptions lead to horizontal
stresses calculated by equation (3).
The picture of rock behavior within the earth’s crust
is not yet complete. The fact that rock is porous and is
filled with a fluid under pressure has an appreciable
influence on its behavior. If a cylinder of rock is
covered with an impermeable jacket, immersed in a
fluid, and then the fluid pressure is increased, the
outer dimensions of the cylinder will decrease. If the
experiment is repeated but with the impermeable
jacket removed (so that the fluid now exerts pressure
throughout the porous structure), then the decrease in
outer dimensions will be much reduced.7 If the material
comprising the rock matrix is quite rigid compared to
the bulk structure, then the elastic and failure behavior
of the bulk structure will be a function of the “effective’’ rock matrix stress (defined by equation 2).
st-p
(2)
ai = effective rock matrix stress in the i direction
Si = total stress in the i direction
p = pore pressure.
The applicability of this equation has been experimentally tested by several investigatorsg-10 and discussed conceptually by others.11-15 In addition to this
effect, some rocks interact with the fluid chemically,
thereby changing their strength.10
Also, one should be aware that fluid moving through
the rock will induce body forces.ll.12.16 An example of
this effect will be mentioned in a later section.
“6
=
Horizontal stresses
(3)
For many cases these assumptions are difficult to
justify. Experiments show the following discrepancies :
(1) Inelastic behavior results during initial compaction
of unconsolidated Sediments; thus, equation (3) does
not indicate the correct value of horizontal stress even
at a condition of zero horizontal Strakz2 (2) Elastic
constants are generally a function of stress level11 and
degree of cementation (geologic age). (3) Those portions of the earth’s crust that are of most interest to the
petroleum industry appear to be significantlyinfluenced
by tectonic movements. (4) For poorly consolidated
rocks at modest depths, a comparison of stress calculated with equations (3) and (4) (using reasonable
values for Poisson’s ratio, angle of internal friction,
etc.) shows that the predicted elastic behavior would
++:>I’
.u -
ASSUMED VALUES:
EARTH STRESSES
In view of these simple concepts of rock behavior,
consider now in situ earth stresses. The total average
vertical stress, at any depth, is equal to the pressure
exerted by the total weight of rock saturated with
fluid (generally water) which lies above it. The densities
of saturated sedimentary rocks do not vary over a
large range, and the average is commonly taken to be
about 1 psi per foot of depth (perhaps kO.1 psi/ft.).
A few investigators17.18 have suggested that this vertical
stress can be diminished by an arching effect. Similarly
Khristianovitch et al.19 have calculated that yielding of
soft Sediments may relieve the vertical stress at the
wellbore. With co&ideration of required arch geo-
.
Y -
P
Sv
0.15
0 . 4 5 p i i l t t . of depth
c‘p
I p a i l f t . of depth
I
I
,
,
I
-’ ,*
.
I
,
I
K
O 0
= o
2
4
6
8
V E R T I C A L S T R E S C . S v . THOUSANDS
10
OF P S I
Fig. 5-Calculated horizontal earth stress
i
I2
18
Rock Mechanics in Oilfield Geology Drilling and Production
actually bring the rock into a state of shear failure. pressure and depth for these high pressure reservoirs.
This comparison is illustrated on Fig. 5.
For the next few paragraphs, two types of reservoirs
Because of these factors, there will be situations will be discussed. First, those reservoirs which exhibit
when equation (3) probably cannot be used to predict the “normal” hydrostatic gradient (0.45 psi per foot of
in situ earth stresses. Rather, a preferable approach depth), and, second, those high pressure reservoirs
might be that of the second school of thought which which exhibit the average pressure to depth relationholds that in situ stresses are significantly influenced ship shown on Fig. 6. Using these pressures and a
by tectonic movements and shear failure of the rock. value of = 30” in equation (4) yields the minimum
Rearrangement of equation (1) yields equation (4), possible horizontal stresses shown on Fig. 7.
giving the minimum physically possible values of
10
horizontal stresses to avoid shear failure.
+
+
-
9;30’
C = U N I T COHESIVE STRENGTH
OF F O R M A T I O N ROCK
S,= I P S I I F T .
J
v 1 w
Formation pressures
~
AVERAGE R E L A T I O N S H I P FOR
H I G H PRESSURE RESERVOIRS
n
5
3
J
L
j
a
To apply this equation, some thought must be given
to the formation pressures likely to be encountered.
Many reservoirs communicate through water either
with the surface or with a free-water table. This water
exerts a hydrostatic pressure at any depth, and the
“normal” original reservoir pressure is considered
equal to the hydrostatic pressure of salt water. The
exact value of the hydrostatic pressure depends on the
density of the salt water, but a common value used is
0.45 psi per foot of depth, representing a brine with
55,000 parts per million of dissolved salt. Many cases
are known, however, where reservoir pressures are less
than or greater than the salt water head.
As a particular example, Cannon and Sullins23 have
reported on many of the high pressure formations
encountered along the Texas and Louisiana Gulf
Coast. Their data are sketched on Fig. 6 together with
a line representing an average relationship between
l2
v1
Y
6-
D E P T H . THOUSANDS O F
‘=ET
Fig. 7-Horizontal stresses predicted by the limit of
shear failure
The question of how the minimum horizontal
stresses are related to actual horizontal stresses can be
approached by considering the stresses arising from
large scale earth movements such as those associated
with faulting (tectonic stresses).
Fig. 8 shows three types of faulting as reported by
van Poollen24 and Hubbert and Willis.15 In both
normal and transcurrent faulting, the minimum stresses
after faulting would be horizontal. In the case of a
normal fault, the minimum horizontal stress would
probably be near those minimum values previously
lL
eæcno
NORMAL FAULTING
6-
J O
k z
-Ge--
t o
THRUST FAULTING
DEPTH, THOUSANDS OF F E E T
Fig. 6-Formation pressure vs. depth for high pressure
formations along Texas and Louisiana Gulf coast
(.from Cannon and S ~ l l i n s ~ ~ )
TRANSCURRENT
FAULTING
Fig. 8-Types
of faulting (from van P ~ o l I e n ~ ~ )
Rock Mechanics in Oilfield Geology Drilling and Production
discussed since it was an approach of horizontal stress
to this limiting value that caused the fault.
For the transcurrent fault, the minimum horizontal
stress could be larger than the minimum values discussed, but could not be larger than the overburden
pressure (if it were larger than the overburden pressure
then a thrust fault would have resulted rather than a
transcurrent fault). In order to have thrust faulting,
the minimum stress must be vertical; both horizontal
stresses would therefore be greater than the overburden pressure.
Along the Texas and Louisiana Gulf Coast, normal
faulting is occurring at the present time. The horizontal
earth stresses in this area should therefore be near the
minimum limiting stress for shear failure which was
previously derived. These same limiting stresses are
probably also near the actual horizontal earth stresses
occurring in the mid-continent regions of the United
States which are characterized by older normal faults.
A tectonic condition of particular interest is the
stress pattern expected around salt domes. Faults
above deep seated domes as well as faults in horizons
above but closer to the salt mass are generally normal
faults. The horizontal stress should therefore be equal
to the minimum values previously discussed. Stresses
in the beds actually pierced by the salt mass, however,
are not so well known. Although normal faults predominate, in some cases thrust faults have been
reported. Therefore, usual stresses should be expected
near salt domes, but with the possibility of an occasional anomalously high stress in zones which flank
salt domes.
79
more plastic shales, compressive strengths are low, and
angles of internal friction probably range from O” to
20”. (The smaller the angle of internal friction, the
more fluid-like is the material.) Therefore, the minimum horizontal stresses in plastic shales are higher
than in more competent rocks as shown on Fig. 9.
For very high stresses, Serata25126and Horvath27
have discussed the transition from “brittle” to “plastic” behavior in competent rocks. Inspection of equations (2) and (4) shows, however, that effective matrix
stresses are only modest even to depths of 10,000 to
15,000 feet, if the formation pressures have not been
depleted. Most fracturing operations occur while formation pressure is relatively high and thus the simple
equations such as (1) and (4) will usually suffice.
FRACTURE INITIATION AND ORIENTATION
The natural state of earth stresses (i.e., vertical stress
generally exceeding horizontal stress) is a condition
which favors vertical hydraulic fractures. Attempts to
control fracture orientation have led to studies of
fracture initiation. Consider first, fracture initiation
from a cylindrical hole such as a wellbore or perforation. If the hole is to be pressured with a fluid, one
must distinguish between a non-penetrating fluid (one
which does not appreciably flow through the rock
during fracture) and a penetrating fluid (one which will
significantly leak into and flow through the rock
during fracture).
Non-penetrating fluid
Plastic behavior
Shales often exhibit different properties from the
more competent sandstones and limestones. For the
w
If non-penetrating fluid is used, it might be supposed that the surrounding rock will behave elastically;
stresses can then be calculated by the Kirsch equations28 shown below.
J
a
iI
9
gr =
a
Y
O
DEPTH,THOUSANDS OF FEET
Fig. 9-Horizontal
stresses in plastic shales
The criterion of failure might be when tangential
stresses are equal to the tensile strength of the rock
(or zero stress if a previous fracture or natural joints
are present). Hubbert15 has shown that in an unequal,
bi-axial stress field, this condition of failure might be
reached even without application of pressure in the
wellbore.
80
Rock Mechanics in OilJeld Geology Drilling and Production
Penetrating fluid
FRACTURE EXTENSION
If fluid flows into the rock during fracture initiation,
then body forces are induced which will influence the
stress pattern. These effects have been estimated by
Cleary11 and Seth,29 the calculations showing that
penetrating fluid will diminish the breakdown pressure
for vertical fractures. Some studies have shown that
use of a penetrating fluid will lead to horizontal fractures. It is envisioned that the fluid pressure will bear
against impermeable boundaries so as to lift the overburden. This effect has been shown experimentally at
low stress levels.17 However, the literature does not
seem to indicate that the same effect can be clearly
demonstrated experimentally with a significantly
unfavorable stress field.
Perhaps a more powerful approach to horizontal
initiation has been the application within recent years
of horizontal notches.30,31 If rock is treated as an
elastic material, then calculations suggest32 that induced stresses near the end of an appropriate notch
would overwhelmingly favor horizontal fractures. At
least one well documented example of a horizontal
fracture initiated from a notch has been recorded in the
literature.33
In view of the non-ideal elastic behavior of rock, it is
not clear that the usual elastic calculations will yield
reliable values for wellbore stresses. Field data suggest,
however, that at least for notches, elastic calculations
will often provide a useful qualitative guide.
In addition to fracture initiation, the mechanics of
fracture extension is also of interest. Sneddon41 has
calculated stresses around a penny-shaped crack in a
perfectly elastic medium. Fig. 10 shows the calculated
stresses in the plane of the crack; at the leading edge of
Effect of anisotropic strength
Other reservoir complexities may significantly influence fracture orientation. Studies of laminated
sedimentary rock@-36 show that rock strength is
typically anisotropic. In the extreme case, particularly
when stresses are low, weaknesses along certain planes
such as bedding planes may control the plane of the
fracture. The effect of existing natural fractures or
joints has been studied by Lamont and Jessen.37 They
found that when stresses were high, the orientation of
a hydraulic fracture is determined primarily by the
orientation of the least-compressive stress around the
extending fracture. A fracture could extend across a
plane of weakness, such as a joint, and, if necessary,
the fracture would turn until it was perpendicular to the
direction of minimum stress. This appears to be in
agreement with the studies of Erdogan and Sih38 and
the studies of Cotterell.39 Castle, Baron, and Habib40
have reported on scaled model studies of hydraulic
fractures. For these homogeneous, isotropic models,
the orientation of the fracture was shown to be perpendicular to the direction of minimum stress.
. :l
2- e
*
+-
t[.,..¡".
I
I
I
l
o. 5
O
I
I
1.0
r
1.5
O
a
Fig. 10-Stresses in the plane of a penny-shaped crack
(as calculated by Sneddon4')
the fracture, infinitely large stresses are induced. These
very large stresses, of course, could not be physically
sustained. Barenblatt and Cherepanov42 have shown
that stresses will remain finite if a region of cohesion is
visualized near the extending edge of the crack. The
concept of cohesion is equivalent to the concept of
surface energy.43 That is, the large stresses induced
near the leading edge of the crack can be relieved by
inelastic processes which absorb energy. The apparent
surface energy is defined as the amount of energy
required to create a unit area of new fracture surface.
Fracture extension pressures are calculated by
making an energy balance. To implement this derivation, visualize that a fracture is inflated and at fracture
extension pressure, P. If a small volume of fluid is
injected so as to increase the crack size slightly, the
work done during injection is balanced against the
increase in elastic strain energy and the increase in
absorbed (surface) energy. One formulation of the
energy balance leads to the Sack44 equation shown as
equation (6).
apparent surface energy
a = radius of the crack
a =
Note that this derivation is consistent with Sneddon's
calculation of crack geometry. For the three-dimen-
Rock Mechanics in Oilfield Geology Drilling and Production
81
sional, penny-shaped crack, the maximum crack width
is given by equation (7).
for several sandstones and limestones are shown in
Table I.
8( 1 - v2)a(P - S )
(7)
TE
It can be shown that Sneddon’s estimate is similar to
Barenblatt’s except near the extending crack tip.
APPARENT SURFACE ENERGIES OF ROCKS45
W =
Rock surface energies
Apparent surface energies of rocks have been
measured by a cleavage technique;45.46 Fig. 11 illustrates the general approach. Blocks of rock are prepared several inches wide, 2 or 3 inches thick, and 1 5 3
feet long. Longitudinal guide slots are cut as well as an
initiating slot at the top. The top of the rock is forced
TABLE I
Rock
Surface
Energy
(in.-lb. Iin.2)
Compressive
Strength
psi
0.78
0.65
0.55
0.054
0.35
0.25
26,750
8,224
29,775
876
8,469
5,490
0.044
0.22
0.1 I
0.24
3,072
17,223
6,922
9,591
Sandstones:
Arizona
Milsap
Colorado
Woodbine
Torpedo
Boise
Limestones:
Austin
Carthage
Leuders
Indiana
~
S E P A R A T I O N A T TOP
FORCE
Under dynamic conditions, it has been observed
that the crack extends in a stepwise, discontinuous
manner as illustrated on Fig. 12. During the “jump”
forward, the crack velocity presumably approaches the
velocity of sound in the rock.47348 Measured values of
apparent surface energy were insensitive to nominal
cleavage rate from about 0.5 to 500 inches/minute.
In another set of experiments, the rock samples were
covered with an impermeable film, submerged in a
low leak-off fluid and cleaved under high pressure.
The fluid leaked into the freshly cleaved surface and
deposited a filter cake in a manner similar to the
-
.
16
-
14
-
NOMINAL CRACK BEHAVIOR
u7
W
3
12-
z
-
1-10I-
ACTUAL STEPWISE
BEHAVIOR
(3
z
Fig. Il-Illustration of cleavage specimen
apart mechanically. The force applied at the top,
width of separation, and crack length are measured.
For a fixed separation at the top, the stored elastic
energy will diminish if the crack extends. On the other
hand, surface energy will be absorbed as the crack
extends. A crack of stable length will be formed when
the change in elastic energy with change in crack
length is just equal to the change in surface energy
with change in crack length. This analysis leads to an
equation explicit in surface energy. Measured values
IJ
Y
2
œ
6-
o
4-
2-
01
O
I
0.2
I
0.4
I
0.6
I
0.8
1
1.0
I
1.2
I
1.4
T I M E , MINUTES
Fig. 12-Example of dynamic fracture propagation in
Carthage limestone46
Rock Mechanics in Oilfield Geology Drilling and Production
82
hydraulic fracturing process. Measured surface energies were sensitive to the external stress, generally
increasing as the applied pressure increased; see
Fig. 13. The apparent surface energies did not appear
to be very sensitive to the type of surrounding fluid used.
20
Y
>z
w œ
w
1.0
+
vertical fracture. The extent of penetration has been
estimated numerically by superposing solutions as
calculated by Schliecher's49 equations. The order of
magnitude of penetration at the wellbore is shown on
Fig. 16. These calculations imply that if bounding
zones are not thick enough, or if pressure in the fracture gets high enough, then the fracture may crack
through into other zones.
6
LIMESTONE
5
O A 0 - 0 1 L BASE FLUID
XO
-WATER BASE FLUID
CONFINING
FRESSURE, P S I
4
Fig. 13-Apparent surface energiesmeasuredunder stress
Restricted vertical fractures
-3
Within the earth, horizontal stresses will not be
equal at every point. The stress, of course, varies with
depth. But more important, an oil bearing zone may
be bounded at top and bottom by zones of higher
horizontal stress such as in shale. If the oil zone is
bounded in this way, then a vertical hydraulic fracture
will not extend evenly in all directions from the wellbore. Rather, it will grow until it reaches the bounding
zones and will then be restricted in vertical growth.
However, it will continue to extend laterally away from
the wellbore. In the actual case, the fracture will extend
up and down for a short distance into the bounding
zones until some equilibrium condition is reached.
Neglecting surface energy effects, the extent of penetration into the bounding zones can be estimated for
simplified cases by applying the Boussinesq equations.
Figs 14 and 15 show a simplified model of a restricted
H O R 1 2 O N T I L S T R E S S I N ZONE 2
P E R P E N D I C U L A R TO P L A N E Of
FRACTURE
U2
-
-
2
x = 0.5
X '
I
1.0
~
>
(
40
30
20
10
50
L
H
Fig. 16-Estimation of the extent of penetration of a
fracture into the bounding zones
Total height of openfracture at the well bore = H (1 x )
P -Bottom hole fracturing pressure
o,-Opposing stress in the producing formation
o,-Opposing stress in the bounding zones
H-Thickness of producing formation
L -Length of fracture
+
ZONE 2
Effect of fluid pressure drop on crack width
Y
ZONE 2
l R O R I Z ö N T I L S T R E S S I N ZONE 1
P E R P E N D I C U L A R TO P L I N E OF
F R A C T U R E Ut
For large hydraulic fractures created under dynamic
conditions, the effect of fluid pressure drop within the
Fig. 14-Simplijied model of restricted vertical fracture
(P-5)
1 1 1 1 1 1 1
~
I
-2.0
~
-1.0
I
~
RELITIVL
Fip. 15-Stress
'
O
,
~
1.0
'
~
~
~
~
~
~
~
~
I
C
T
~
~
2.0
POSITION
svstem eauivalent to restricted fracture
Fir. 17-Sketch
o f restricted vertical fracture
Rock Mechanics in Oilfield Gt:ology Drilling and Production
fracture will have an effect on fracture width. Take the
case of a restricted vertical fracture, shown on Fig. 17,
as an example. If the pressure near the extending edge
of the crack is consistent with energy requirements
[i.e., eqn. (6)], then additional crack width must result
from dynamic fluid pressure drop within the fracture.
Following the approach outlined in reference50 if fluid
is in laminar flow, the maximum crack width at the
wellbore is given approximately by equation (8).
+ 3.2 x lo3(
where L is shown on Fig. 17.
For the two-dimensional crack, Sneddon and
Elliott51have given the relation shown by equation (9).
W=
2( I - v2) (P- S)H
E
(9)
Combining equations (8) and (9) suggests that for
restricted vertical fractures, pressures in the fracture
will exceed earth stresses by a few tens of psi (for tall
cracks) to perhaps 1000 psi (for thin zones). Combining these estimates with those for earth stresses gives
the estimates for bottom-hole fracturing pressures
shown on Fig. 18. This figure also shows bottom-hole
fracturing pressures calculated from actual field
treatments. For most of these cases, the pressures feil
in the range predicted for vertical fractures.
8-
6O
D E P T H . (THOUSANDS O F F E E T 1
Fig. 18-Theoretically predicted bottom-holefracturing
pressures and$eId data”
CONCLUSIONS
1. Three aspects of rock mechanics have been of
value in developing hydraulic fracturing theories,
viz., elastic behavior, shear failure criteria, and
concepts of fracture propagation.
2. These ideas of rock behavior lead to estimates of
83
horizontal earth stresses less than estimates of
vertical earth stresses. Such a condition favors
vertical fractures.
3. Attempts to orient fractures have involved use of
penetrating-fluid pressures and/or stress concentrating notches.
4. Studies indicate, however, that if stresses are
sufficiently large compared to rock strength,
fractures will deviate if necessary and approach,
an orientation perpendicular to the direction of
least stress. Existing natural fractures or joints
will not necessarily prevent this orientation.
5. Pressures needed to fracture must be consistent
with energy equations such that sufficient surface
energy is available during crack propagation.
6. For long fractures, pressures must also be consistent with fluid dynamics. Increased pressure
drop in the fracture will lead to increased fracture
width.
Nomenclature
a = radius of a crack
C = unit cohesive strength
E = Young’s modulus
H = height of a restricted vertical fracture
L = length of a restricted vertical fracture
p = pore pressure
P = pressure in a fracture
r = radius
rh = radius of a hole
S = total opposing stress
St = total stress in the i direction
Sh = total horizontal earth stress
S, = total vertical earth stress
W = maximum width of a crack
U
= apparent surface energy
û = angle from the axis of stress ua
v = Poisson’s ratio
ua = a uniaxial compressive stress
05
= effective rock matrix stress in the i direction
un = stress normal to the plane of failure
ur = radial stress
U@
= tangential stress
T
= shear stress
4 = angle of internal friction
References
1 . Theory of Flow and Fracture of Solids. A. NADAI. New
York, McGraw-Hill Book Co., Vol. 1, 2nd edn., 1950.
2. CARPENTER, C . B. and SPENCER, G. B., U.S. Dept. of
Interior, Bureau of Mines, RI.3540, 1940.
3. DERESIEWTCZ, H., Appl. Mech. Rev., 1958, 11,259-61.
Rock Mechanics in Oilfield Geology Drilling and Production
84
4. BRANDT, H., J. Appl. Mech., 1955, 22, 479-86.
5. NISHIHARA. M., The Doshisha Ennineerinn Review. 1957.
8 (2), 32-54.
6. BOTSET, H. G. and REED, D. W., Bull. Amer. Ass.
Petrol. Geol., 1935, 19, 1053-1060.
7. GEERTSMA, J., Trans. AIME, 1957, 210, 331-340.
8. McHENRY, D., Troisieme Congres Des Grands Barrages,
R-48,1948
9. LELIAVSKY, S., Proc. Amer. Soc. Civil Engrs., 1945, 71,
1474-1517.
10. BARON, G., CASTEL, Y., and HABIB, P., Reue D e
L‘Institut Francais du Pétrole, 1963, 78-96.
11. CLEARY, J. M., Illinois State Geological Survey, Circulars
251, 252, and 281, 1958, 1959.
12. LUBINSKI, A., Proc. Second US.Natl. Cong. Appl. Mech.,
1954,247-256.
13. BRAHTZ, J. H. A., Second Congress on Large Dams,
Washington, D.C., 1936, Communication 1,43-71.
14. TERYAGHI, K., ASTM P ~ O C1945,
.,
45,777-791.
15. HUBBERT, M. K. and WILLIS, D. G., Trans. AIME,
1957,210, 153-166.
16. Theory of Elasticity, TIMOSHENKO, S. and GOODIER,
J. N., New York, McGraw-Hill Book Co., 2nd edn., 1951.
17. SCOTT, P. P., Jr., BEARDEN, W. G., and HOWARD,
G. C., Trans. AIME, 1953, 198, 111-124.
18. HOWARD, G. C. and FAST, C. R., Trans. AIME, 1950,
189 -51-64
- - ..
19. KHRISTIANOVITCH, S. A., ZELTOV, Y. P., BARENBLATT, G. I., and MAXIMOVITCH, G. K., Proc. Fifth
WorldPet. Cona.. N.Y.. 1959. Section II. 289-295.
20. HARRISON, E.’, KIÉSCHNICK, W.’F., Jr., and McGUIRE, W. J., Trum. AIME, 1954,201,252-263.
21. CRITTENDON, B. C., Journ. Pet. Tech., 1959, 11 (lo),
21-29.
22. HENDRON, A. J., Jr., “The Behavior of Sand in OneDimensional Compression”, Ph.D. Thesis, University of
Illinois, 1963.
23. CANNON, E. and SULLINS, R. S., Oil and Gas Journal,
1946. 45 (3). 120. 123. 124.
24. van POOLLEN, H. K., Quart. of Colorado School of Mines,
1957, 52 (3), 115-127.
25. SERATA, S., Fourth Symposium on Rock Mechanics,
Pennsylvania State University, 1961, 73-82.
26. SERATA, S., Sixth Symposium on Rock Mechanics,
University of Missouri at Rolla, 1964, 260-292.
- - - y
27. HORVATH, J., Int. J. Rock Mech. Min.Sci., 1965, 2, 327340.
28. MILES, A. J. and TOPPING, A. D., Trans. AIME, 1949,
179, 186191.
29. SETH, M. S., “Stresses and Displacements in Transversely
Isotropic, Porous, Elastic Media Under Transient Pore
Pressure,” Ph.D. Dissertation, University of Texas, 1966.
30. BROWN, R. W. and LOPER, J. L., Trans. AIME, 1961,
222,483488.
31. PITTMAN, F. C., HARRIMAN, D. W., and ST. JOHN,
J. C., Trans. AIME, 1961, 222,489495.
32. Theory of Notch Stresses, H. NEUBER, Ann Arbor,
Michigan, Edwards, 1946.
33. REYNOLDS, J. J., SCOTT, J. B., POPHAM, J. L., and
COFFER, H. F., Tvans. AIME, 1961,222,371-376.
34. DONATH. F. A.. G.S.A. Bull.. 1961. 72. 985-990.
35. CHENEVERT, M. E. and GÁTLIN, C., Trans. AIME,
Part II, 1965, 234, 66-77.
36. HOBBS, D. W., Int. J. Rock Mech. Min. Sci., 1964, 1,
785-796
I_-.
-I-
37. LAMONT, N. and JESSEN, F. W., Trans. AIME, Part I,
1963,228, 203-209.
38. ERDOGAN, F. and SIH, G. C., Trans. ASME, Jour. of
Basic Engineering, 1963, 85, Series D, 519-527.
39. COTTERELL, B., Int. Jour. Frac. Mech., 1965, 1 (2), 66103.
40. CASTLE, Y.,BARON, G., and HABIB, P., Comptes
Rendus du Deuxieme Colloque de L‘Association de
Recherche Sur Les Techniques De Forage Et De Production. 1965. 167-186.
41. SNEDDON, Í.N., Proc. Roy. Soc., 1946, A, 187,229-260.
42. BARENBLATT, G. I. and CHEREPANOV, G. P., PMM,
1961, 25 (4), 752-753.
43. BARENBLATT, G. I., PMM, 1959,23 (9,893-900.
44. SACK. R. A.. Proc. Phvs. Soc. ofLondon. 1946.58.729-736.
45. PERKINS, T. K. and BARTLETT, L.’E., Trais. AIME,
Part II, 1963, 228, 307-313.
46. PERKINS, T. K. and KRECH, W. W., Soc. Pet. Eng.
paper 1540, presented Dallas, Texas, October, 1966.
47. KOSTROV, B. V., PMM, 1964, 28 (4), 644-652.
48. MANSINHA, L., J . Mech. Phys. Solids, 1964, 12, 353-360.
49. SCHLEICHER, F., Der Bauingenieur, 1926,48,931-935.
50. PERKINS, T. K. and KERN, L. R., Trans. AIME, Part I,
1961, 222, 937-949.
51. SNEDDON, I. N. and ELLIOTT, H. A., Quart. of AppL
Math., 1964, 4, 262-267.
FRACTURING THEORIES
Abstract
Three aspects of rock mechanics have been of great
value in developing hydraulic fracturing theories, viz. :
(1) Elastic behavior. (2) Shear failure criteria. (3) Concepts of fracture propagation.
Within the framework of these concepts, the following topics will be discussed: (i) The relationship
between rock matrix stress and total stress. (2) Estimates and significance of in situ earth stresses. (3) Factors influencing fracture orientation. (4) Fracture
extension pressures. ( 5 ) Apparent rock surface energies. (6) Fracture widths.
Résumé
Trois aspects de la mécanique de la roche ont largement contribué au développement des théories de la
fracturation causée par une force hydraulique. . .
(poussées d’eau) . c.à.d.: (i) Le comportement
élastique. (2) Les critères des efforts de cisaillement qui
causent la faille. (3) Les concepts de l’extension de la
fracture.
C‘est dans le cadre de ces concepts que les sujets qui
suivent vont être discutés: (1) Les rapports entre les
efforts de la matrice de la roche et la somme des efforts
(2) Evaluation et signification des efforts terrestres sur
les lieux (“in situ”). (3) Les facteurs qui influencent
l’orientation de la fracture. (4) Les pressions qui
résultent de l’extension de la fracture. (5) Les energies
évidentes à la surface de la roche. (6) Les largeurs de la
fracture.
INTRODUCTION
departure from idealized linear behavior, and for many
problems the linear elastic theory must be augmented
with additional concepts or approximations.
.
One of the significant advances in petroleum technology within the past twenty years has been the
development of hydraulic fracturing processes. Within
this same period, understanding of the mechanics of
Limits to idealized behavior
rock behavior has increased. It is the purpose of this
paper to show how knowledge of rock mechanics has
There is clearly a limit to elastic behavior if a
been applied in the understanding and further develop- sufficiently large uniaxial tensile stress is applied (the
ment of hydraulic fracturing processes.
tensile strength). Also, if a combination of stresses is
Those aspects of rock behavior that will be of interest applied, there is a limit beyond which shear failure will
are briefly summarized, and then in situ earth stresses occur. For present purposes, it is sufficient to describe
and factors influencing hydraulic fracture initiation this envelope of shear failure by the well known Mohr
and extension are discussed.
failure theory.’ If stresses are not too large, the shear
limit can be written in a simple manner as shown by
equation (i).
BEHAVIOR OF ROCKS
r = C‘+un tan q5
(1)
= shear stress at failure
Many investigators have treated rocks as linear,
C = unit cohesive strength
elastic materials. Such an assumption permits the
powerful mathematics of the linear elastic theory to be
un = stress normal to the plane of failure
applied in solving problems. For many circumstances,
4 = angle of internal friction.
this approach appears to be entirely justified, and
meaningful solutions can be obtained. On the other Within the region of shear stability, the rock behavior
hand, studies of rock behavior generally show a is presumably elastic. However, studies show at least
four additional complexities :
(i) For rocks of low to moderate strength, there is
a pronounced hysteresis for moderate stress changes.
by T. K. PERKINS, Atlantic Richfield Co., U.S.A.
15
Rock Mechanics in Oilfield Geology Drilling and Production
76
12
~~
-- -z
<n
REDUCTION
IN PORE SPACE, PER C E N T
Fig. 1-Compressibility behavior of Bartlesville
sandstone (from Carpenter and Spencer2)
Fig. 1 illustrates this type of behavior with data of
Carpenter and Spencer2 showing the compressibility
of Bartlesville sandstone. The hysteresis obviously
indicates other than purely elastic behavior.
(2) For large stress changes, elasticity of granular
materials is at best non-linear. See Deresiewicz3 for a
review of mechanical behavior of granular media. As
an illustration, Brandt4 has computed the volumepressure relationship for an aggregate by applying the
Hertz theory for the deformation of elastic spheres.
Fig. 2 shows experimental data together with Brandt’s
computed curves.
1.0
0.90
0.96
k
0
8-
fn
m y w>
o
W n .a
œ
g3 Eo
-
4-
bfn
<n
.~
a -0
+
z
w
o
u)
z
w -4
(4) When large stresses are first applied to unconsolidated materials, there is generally a distinct
“compaction” behavior which is different from subsequent “elastic” behavior. Fig. 4 shows the data of
Botset and Reed6 which illustrates an initial severe
compaction of unconsolidated sand. Repeated cycles
0.94
POROSITY A T PRESSURE
POROSITY A T ZERO PRESSURE
Fig. 2-Porosity
variation as a function of pressure
(from Brandt4)
Curve A-Basal Tuscaloosa, Miss. ;Porosity 15 %
Curve B-Basal Tuscaloosa, Miss. ;Porosity 24 %
Curves C & D-Theoretical Curve
Curve E-Southern California Coast; Porosity 25 %
Curve F-Southern California Coast; Porosity 22
35
36
37
38
39
40
41
42
POROSITY. PER C E N T
Fig. 4-Compaction of unconsolidated sand
(from Botset and Reed6)
II
Rock Mechanics in Oilfield Geology Drilling and Prcduction
of loading show continued crushing as well as “elastic”
behavi or.
Even though rock behavior is complex, for certain
stress ranges and for certain kinds of problems, application of the linear elastic theory will given meaningful
engineering solutions.
metry, however, it seems likely that these various
effects would be limited to a region close by the wellbore, and that generally in the reservoir the weight of
the full overburden would bear.
Effect of interstitial fluid pressure
There are two broad schools of thought concerning
the prediction of in situ horizontal earth stresses. In
effect, with one approach it is supposed that the
environment and state of earth stress are known from
the time of initial mineral deposition. In using this
approach,l7J0.21 it is commonly assumed that behavior is purely elastic, that elastic constants were not a
function of stress (depth) or age (degree of cementation) and that there were no tectonic influences (a condition of zero horizontal strain is generally assumed).
If grain compressibility is small compared to bulk
compressibility, these assumptions lead to horizontal
stresses calculated by equation (3).
The picture of rock behavior within the earth’s crust
is not yet complete. The fact that rock is porous and is
filled with a fluid under pressure has an appreciable
influence on its behavior. If a cylinder of rock is
covered with an impermeable jacket, immersed in a
fluid, and then the fluid pressure is increased, the
outer dimensions of the cylinder will decrease. If the
experiment is repeated but with the impermeable
jacket removed (so that the fluid now exerts pressure
throughout the porous structure), then the decrease in
outer dimensions will be much reduced.7 If the material
comprising the rock matrix is quite rigid compared to
the bulk structure, then the elastic and failure behavior
of the bulk structure will be a function of the “effective’’ rock matrix stress (defined by equation 2).
st-p
(2)
ai = effective rock matrix stress in the i direction
Si = total stress in the i direction
p = pore pressure.
The applicability of this equation has been experimentally tested by several investigatorsg-10 and discussed conceptually by others.11-15 In addition to this
effect, some rocks interact with the fluid chemically,
thereby changing their strength.10
Also, one should be aware that fluid moving through
the rock will induce body forces.ll.12.16 An example of
this effect will be mentioned in a later section.
“6
=
Horizontal stresses
(3)
For many cases these assumptions are difficult to
justify. Experiments show the following discrepancies :
(1) Inelastic behavior results during initial compaction
of unconsolidated Sediments; thus, equation (3) does
not indicate the correct value of horizontal stress even
at a condition of zero horizontal Strakz2 (2) Elastic
constants are generally a function of stress level11 and
degree of cementation (geologic age). (3) Those portions of the earth’s crust that are of most interest to the
petroleum industry appear to be significantlyinfluenced
by tectonic movements. (4) For poorly consolidated
rocks at modest depths, a comparison of stress calculated with equations (3) and (4) (using reasonable
values for Poisson’s ratio, angle of internal friction,
etc.) shows that the predicted elastic behavior would
++:>I’
.u -
ASSUMED VALUES:
EARTH STRESSES
In view of these simple concepts of rock behavior,
consider now in situ earth stresses. The total average
vertical stress, at any depth, is equal to the pressure
exerted by the total weight of rock saturated with
fluid (generally water) which lies above it. The densities
of saturated sedimentary rocks do not vary over a
large range, and the average is commonly taken to be
about 1 psi per foot of depth (perhaps kO.1 psi/ft.).
A few investigators17.18 have suggested that this vertical
stress can be diminished by an arching effect. Similarly
Khristianovitch et al.19 have calculated that yielding of
soft Sediments may relieve the vertical stress at the
wellbore. With co&ideration of required arch geo-
.
Y -
P
Sv
0.15
0 . 4 5 p i i l t t . of depth
c‘p
I p a i l f t . of depth
I
I
,
,
I
-’ ,*
.
I
,
I
K
O 0
= o
2
4
6
8
V E R T I C A L S T R E S C . S v . THOUSANDS
10
OF P S I
Fig. 5-Calculated horizontal earth stress
i
I2
18
Rock Mechanics in Oilfield Geology Drilling and Production
actually bring the rock into a state of shear failure. pressure and depth for these high pressure reservoirs.
This comparison is illustrated on Fig. 5.
For the next few paragraphs, two types of reservoirs
Because of these factors, there will be situations will be discussed. First, those reservoirs which exhibit
when equation (3) probably cannot be used to predict the “normal” hydrostatic gradient (0.45 psi per foot of
in situ earth stresses. Rather, a preferable approach depth), and, second, those high pressure reservoirs
might be that of the second school of thought which which exhibit the average pressure to depth relationholds that in situ stresses are significantly influenced ship shown on Fig. 6. Using these pressures and a
by tectonic movements and shear failure of the rock. value of = 30” in equation (4) yields the minimum
Rearrangement of equation (1) yields equation (4), possible horizontal stresses shown on Fig. 7.
giving the minimum physically possible values of
10
horizontal stresses to avoid shear failure.
+
+
-
9;30’
C = U N I T COHESIVE STRENGTH
OF F O R M A T I O N ROCK
S,= I P S I I F T .
J
v 1 w
Formation pressures
~
AVERAGE R E L A T I O N S H I P FOR
H I G H PRESSURE RESERVOIRS
n
5
3
J
L
j
a
To apply this equation, some thought must be given
to the formation pressures likely to be encountered.
Many reservoirs communicate through water either
with the surface or with a free-water table. This water
exerts a hydrostatic pressure at any depth, and the
“normal” original reservoir pressure is considered
equal to the hydrostatic pressure of salt water. The
exact value of the hydrostatic pressure depends on the
density of the salt water, but a common value used is
0.45 psi per foot of depth, representing a brine with
55,000 parts per million of dissolved salt. Many cases
are known, however, where reservoir pressures are less
than or greater than the salt water head.
As a particular example, Cannon and Sullins23 have
reported on many of the high pressure formations
encountered along the Texas and Louisiana Gulf
Coast. Their data are sketched on Fig. 6 together with
a line representing an average relationship between
l2
v1
Y
6-
D E P T H . THOUSANDS O F
‘=ET
Fig. 7-Horizontal stresses predicted by the limit of
shear failure
The question of how the minimum horizontal
stresses are related to actual horizontal stresses can be
approached by considering the stresses arising from
large scale earth movements such as those associated
with faulting (tectonic stresses).
Fig. 8 shows three types of faulting as reported by
van Poollen24 and Hubbert and Willis.15 In both
normal and transcurrent faulting, the minimum stresses
after faulting would be horizontal. In the case of a
normal fault, the minimum horizontal stress would
probably be near those minimum values previously
lL
eæcno
NORMAL FAULTING
6-
J O
k z
-Ge--
t o
THRUST FAULTING
DEPTH, THOUSANDS OF F E E T
Fig. 6-Formation pressure vs. depth for high pressure
formations along Texas and Louisiana Gulf coast
(.from Cannon and S ~ l l i n s ~ ~ )
TRANSCURRENT
FAULTING
Fig. 8-Types
of faulting (from van P ~ o l I e n ~ ~ )
Rock Mechanics in Oilfield Geology Drilling and Production
discussed since it was an approach of horizontal stress
to this limiting value that caused the fault.
For the transcurrent fault, the minimum horizontal
stress could be larger than the minimum values discussed, but could not be larger than the overburden
pressure (if it were larger than the overburden pressure
then a thrust fault would have resulted rather than a
transcurrent fault). In order to have thrust faulting,
the minimum stress must be vertical; both horizontal
stresses would therefore be greater than the overburden pressure.
Along the Texas and Louisiana Gulf Coast, normal
faulting is occurring at the present time. The horizontal
earth stresses in this area should therefore be near the
minimum limiting stress for shear failure which was
previously derived. These same limiting stresses are
probably also near the actual horizontal earth stresses
occurring in the mid-continent regions of the United
States which are characterized by older normal faults.
A tectonic condition of particular interest is the
stress pattern expected around salt domes. Faults
above deep seated domes as well as faults in horizons
above but closer to the salt mass are generally normal
faults. The horizontal stress should therefore be equal
to the minimum values previously discussed. Stresses
in the beds actually pierced by the salt mass, however,
are not so well known. Although normal faults predominate, in some cases thrust faults have been
reported. Therefore, usual stresses should be expected
near salt domes, but with the possibility of an occasional anomalously high stress in zones which flank
salt domes.
79
more plastic shales, compressive strengths are low, and
angles of internal friction probably range from O” to
20”. (The smaller the angle of internal friction, the
more fluid-like is the material.) Therefore, the minimum horizontal stresses in plastic shales are higher
than in more competent rocks as shown on Fig. 9.
For very high stresses, Serata25126and Horvath27
have discussed the transition from “brittle” to “plastic” behavior in competent rocks. Inspection of equations (2) and (4) shows, however, that effective matrix
stresses are only modest even to depths of 10,000 to
15,000 feet, if the formation pressures have not been
depleted. Most fracturing operations occur while formation pressure is relatively high and thus the simple
equations such as (1) and (4) will usually suffice.
FRACTURE INITIATION AND ORIENTATION
The natural state of earth stresses (i.e., vertical stress
generally exceeding horizontal stress) is a condition
which favors vertical hydraulic fractures. Attempts to
control fracture orientation have led to studies of
fracture initiation. Consider first, fracture initiation
from a cylindrical hole such as a wellbore or perforation. If the hole is to be pressured with a fluid, one
must distinguish between a non-penetrating fluid (one
which does not appreciably flow through the rock
during fracture) and a penetrating fluid (one which will
significantly leak into and flow through the rock
during fracture).
Non-penetrating fluid
Plastic behavior
Shales often exhibit different properties from the
more competent sandstones and limestones. For the
w
If non-penetrating fluid is used, it might be supposed that the surrounding rock will behave elastically;
stresses can then be calculated by the Kirsch equations28 shown below.
J
a
iI
9
gr =
a
Y
O
DEPTH,THOUSANDS OF FEET
Fig. 9-Horizontal
stresses in plastic shales
The criterion of failure might be when tangential
stresses are equal to the tensile strength of the rock
(or zero stress if a previous fracture or natural joints
are present). Hubbert15 has shown that in an unequal,
bi-axial stress field, this condition of failure might be
reached even without application of pressure in the
wellbore.
80
Rock Mechanics in OilJeld Geology Drilling and Production
Penetrating fluid
FRACTURE EXTENSION
If fluid flows into the rock during fracture initiation,
then body forces are induced which will influence the
stress pattern. These effects have been estimated by
Cleary11 and Seth,29 the calculations showing that
penetrating fluid will diminish the breakdown pressure
for vertical fractures. Some studies have shown that
use of a penetrating fluid will lead to horizontal fractures. It is envisioned that the fluid pressure will bear
against impermeable boundaries so as to lift the overburden. This effect has been shown experimentally at
low stress levels.17 However, the literature does not
seem to indicate that the same effect can be clearly
demonstrated experimentally with a significantly
unfavorable stress field.
Perhaps a more powerful approach to horizontal
initiation has been the application within recent years
of horizontal notches.30,31 If rock is treated as an
elastic material, then calculations suggest32 that induced stresses near the end of an appropriate notch
would overwhelmingly favor horizontal fractures. At
least one well documented example of a horizontal
fracture initiated from a notch has been recorded in the
literature.33
In view of the non-ideal elastic behavior of rock, it is
not clear that the usual elastic calculations will yield
reliable values for wellbore stresses. Field data suggest,
however, that at least for notches, elastic calculations
will often provide a useful qualitative guide.
In addition to fracture initiation, the mechanics of
fracture extension is also of interest. Sneddon41 has
calculated stresses around a penny-shaped crack in a
perfectly elastic medium. Fig. 10 shows the calculated
stresses in the plane of the crack; at the leading edge of
Effect of anisotropic strength
Other reservoir complexities may significantly influence fracture orientation. Studies of laminated
sedimentary rock@-36 show that rock strength is
typically anisotropic. In the extreme case, particularly
when stresses are low, weaknesses along certain planes
such as bedding planes may control the plane of the
fracture. The effect of existing natural fractures or
joints has been studied by Lamont and Jessen.37 They
found that when stresses were high, the orientation of
a hydraulic fracture is determined primarily by the
orientation of the least-compressive stress around the
extending fracture. A fracture could extend across a
plane of weakness, such as a joint, and, if necessary,
the fracture would turn until it was perpendicular to the
direction of minimum stress. This appears to be in
agreement with the studies of Erdogan and Sih38 and
the studies of Cotterell.39 Castle, Baron, and Habib40
have reported on scaled model studies of hydraulic
fractures. For these homogeneous, isotropic models,
the orientation of the fracture was shown to be perpendicular to the direction of minimum stress.
. :l
2- e
*
+-
t[.,..¡".
I
I
I
l
o. 5
O
I
I
1.0
r
1.5
O
a
Fig. 10-Stresses in the plane of a penny-shaped crack
(as calculated by Sneddon4')
the fracture, infinitely large stresses are induced. These
very large stresses, of course, could not be physically
sustained. Barenblatt and Cherepanov42 have shown
that stresses will remain finite if a region of cohesion is
visualized near the extending edge of the crack. The
concept of cohesion is equivalent to the concept of
surface energy.43 That is, the large stresses induced
near the leading edge of the crack can be relieved by
inelastic processes which absorb energy. The apparent
surface energy is defined as the amount of energy
required to create a unit area of new fracture surface.
Fracture extension pressures are calculated by
making an energy balance. To implement this derivation, visualize that a fracture is inflated and at fracture
extension pressure, P. If a small volume of fluid is
injected so as to increase the crack size slightly, the
work done during injection is balanced against the
increase in elastic strain energy and the increase in
absorbed (surface) energy. One formulation of the
energy balance leads to the Sack44 equation shown as
equation (6).
apparent surface energy
a = radius of the crack
a =
Note that this derivation is consistent with Sneddon's
calculation of crack geometry. For the three-dimen-
Rock Mechanics in Oilfield Geology Drilling and Production
81
sional, penny-shaped crack, the maximum crack width
is given by equation (7).
for several sandstones and limestones are shown in
Table I.
8( 1 - v2)a(P - S )
(7)
TE
It can be shown that Sneddon’s estimate is similar to
Barenblatt’s except near the extending crack tip.
APPARENT SURFACE ENERGIES OF ROCKS45
W =
Rock surface energies
Apparent surface energies of rocks have been
measured by a cleavage technique;45.46 Fig. 11 illustrates the general approach. Blocks of rock are prepared several inches wide, 2 or 3 inches thick, and 1 5 3
feet long. Longitudinal guide slots are cut as well as an
initiating slot at the top. The top of the rock is forced
TABLE I
Rock
Surface
Energy
(in.-lb. Iin.2)
Compressive
Strength
psi
0.78
0.65
0.55
0.054
0.35
0.25
26,750
8,224
29,775
876
8,469
5,490
0.044
0.22
0.1 I
0.24
3,072
17,223
6,922
9,591
Sandstones:
Arizona
Milsap
Colorado
Woodbine
Torpedo
Boise
Limestones:
Austin
Carthage
Leuders
Indiana
~
S E P A R A T I O N A T TOP
FORCE
Under dynamic conditions, it has been observed
that the crack extends in a stepwise, discontinuous
manner as illustrated on Fig. 12. During the “jump”
forward, the crack velocity presumably approaches the
velocity of sound in the rock.47348 Measured values of
apparent surface energy were insensitive to nominal
cleavage rate from about 0.5 to 500 inches/minute.
In another set of experiments, the rock samples were
covered with an impermeable film, submerged in a
low leak-off fluid and cleaved under high pressure.
The fluid leaked into the freshly cleaved surface and
deposited a filter cake in a manner similar to the
-
.
16
-
14
-
NOMINAL CRACK BEHAVIOR
u7
W
3
12-
z
-
1-10I-
ACTUAL STEPWISE
BEHAVIOR
(3
z
Fig. Il-Illustration of cleavage specimen
apart mechanically. The force applied at the top,
width of separation, and crack length are measured.
For a fixed separation at the top, the stored elastic
energy will diminish if the crack extends. On the other
hand, surface energy will be absorbed as the crack
extends. A crack of stable length will be formed when
the change in elastic energy with change in crack
length is just equal to the change in surface energy
with change in crack length. This analysis leads to an
equation explicit in surface energy. Measured values
IJ
Y
2
œ
6-
o
4-
2-
01
O
I
0.2
I
0.4
I
0.6
I
0.8
1
1.0
I
1.2
I
1.4
T I M E , MINUTES
Fig. 12-Example of dynamic fracture propagation in
Carthage limestone46
Rock Mechanics in Oilfield Geology Drilling and Production
82
hydraulic fracturing process. Measured surface energies were sensitive to the external stress, generally
increasing as the applied pressure increased; see
Fig. 13. The apparent surface energies did not appear
to be very sensitive to the type of surrounding fluid used.
20
Y
>z
w œ
w
1.0
+
vertical fracture. The extent of penetration has been
estimated numerically by superposing solutions as
calculated by Schliecher's49 equations. The order of
magnitude of penetration at the wellbore is shown on
Fig. 16. These calculations imply that if bounding
zones are not thick enough, or if pressure in the fracture gets high enough, then the fracture may crack
through into other zones.
6
LIMESTONE
5
O A 0 - 0 1 L BASE FLUID
XO
-WATER BASE FLUID
CONFINING
FRESSURE, P S I
4
Fig. 13-Apparent surface energiesmeasuredunder stress
Restricted vertical fractures
-3
Within the earth, horizontal stresses will not be
equal at every point. The stress, of course, varies with
depth. But more important, an oil bearing zone may
be bounded at top and bottom by zones of higher
horizontal stress such as in shale. If the oil zone is
bounded in this way, then a vertical hydraulic fracture
will not extend evenly in all directions from the wellbore. Rather, it will grow until it reaches the bounding
zones and will then be restricted in vertical growth.
However, it will continue to extend laterally away from
the wellbore. In the actual case, the fracture will extend
up and down for a short distance into the bounding
zones until some equilibrium condition is reached.
Neglecting surface energy effects, the extent of penetration into the bounding zones can be estimated for
simplified cases by applying the Boussinesq equations.
Figs 14 and 15 show a simplified model of a restricted
H O R 1 2 O N T I L S T R E S S I N ZONE 2
P E R P E N D I C U L A R TO P L A N E Of
FRACTURE
U2
-
-
2
x = 0.5
X '
I
1.0
~
>
(
40
30
20
10
50
L
H
Fig. 16-Estimation of the extent of penetration of a
fracture into the bounding zones
Total height of openfracture at the well bore = H (1 x )
P -Bottom hole fracturing pressure
o,-Opposing stress in the producing formation
o,-Opposing stress in the bounding zones
H-Thickness of producing formation
L -Length of fracture
+
ZONE 2
Effect of fluid pressure drop on crack width
Y
ZONE 2
l R O R I Z ö N T I L S T R E S S I N ZONE 1
P E R P E N D I C U L A R TO P L I N E OF
F R A C T U R E Ut
For large hydraulic fractures created under dynamic
conditions, the effect of fluid pressure drop within the
Fig. 14-Simplijied model of restricted vertical fracture
(P-5)
1 1 1 1 1 1 1
~
I
-2.0
~
-1.0
I
~
RELITIVL
Fip. 15-Stress
'
O
,
~
1.0
'
~
~
~
~
~
~
~
~
I
C
T
~
~
2.0
POSITION
svstem eauivalent to restricted fracture
Fir. 17-Sketch
o f restricted vertical fracture
Rock Mechanics in Oilfield Gt:ology Drilling and Production
fracture will have an effect on fracture width. Take the
case of a restricted vertical fracture, shown on Fig. 17,
as an example. If the pressure near the extending edge
of the crack is consistent with energy requirements
[i.e., eqn. (6)], then additional crack width must result
from dynamic fluid pressure drop within the fracture.
Following the approach outlined in reference50 if fluid
is in laminar flow, the maximum crack width at the
wellbore is given approximately by equation (8).
+ 3.2 x lo3(
where L is shown on Fig. 17.
For the two-dimensional crack, Sneddon and
Elliott51have given the relation shown by equation (9).
W=
2( I - v2) (P- S)H
E
(9)
Combining equations (8) and (9) suggests that for
restricted vertical fractures, pressures in the fracture
will exceed earth stresses by a few tens of psi (for tall
cracks) to perhaps 1000 psi (for thin zones). Combining these estimates with those for earth stresses gives
the estimates for bottom-hole fracturing pressures
shown on Fig. 18. This figure also shows bottom-hole
fracturing pressures calculated from actual field
treatments. For most of these cases, the pressures feil
in the range predicted for vertical fractures.
8-
6O
D E P T H . (THOUSANDS O F F E E T 1
Fig. 18-Theoretically predicted bottom-holefracturing
pressures and$eId data”
CONCLUSIONS
1. Three aspects of rock mechanics have been of
value in developing hydraulic fracturing theories,
viz., elastic behavior, shear failure criteria, and
concepts of fracture propagation.
2. These ideas of rock behavior lead to estimates of
83
horizontal earth stresses less than estimates of
vertical earth stresses. Such a condition favors
vertical fractures.
3. Attempts to orient fractures have involved use of
penetrating-fluid pressures and/or stress concentrating notches.
4. Studies indicate, however, that if stresses are
sufficiently large compared to rock strength,
fractures will deviate if necessary and approach,
an orientation perpendicular to the direction of
least stress. Existing natural fractures or joints
will not necessarily prevent this orientation.
5. Pressures needed to fracture must be consistent
with energy equations such that sufficient surface
energy is available during crack propagation.
6. For long fractures, pressures must also be consistent with fluid dynamics. Increased pressure
drop in the fracture will lead to increased fracture
width.
Nomenclature
a = radius of a crack
C = unit cohesive strength
E = Young’s modulus
H = height of a restricted vertical fracture
L = length of a restricted vertical fracture
p = pore pressure
P = pressure in a fracture
r = radius
rh = radius of a hole
S = total opposing stress
St = total stress in the i direction
Sh = total horizontal earth stress
S, = total vertical earth stress
W = maximum width of a crack
U
= apparent surface energy
û = angle from the axis of stress ua
v = Poisson’s ratio
ua = a uniaxial compressive stress
05
= effective rock matrix stress in the i direction
un = stress normal to the plane of failure
ur = radial stress
U@
= tangential stress
T
= shear stress
4 = angle of internal friction
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Rock Mechanics in Oilfield Geology Drilling and Production
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