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Transcript of The Effects of in-situ Stresses and Layer Properties on the Containment of a Hydraulic...
THE EFFECTS OF IN-SITU STRESSES
AND LAYER PROPERTIES ON THE CONTAINMENT OF
A HYDRAULIC FRACTURE
BY
C. H. YE~'J
PROFESSOR
and
Y.J. CHIOU
GRADUATE STUDENT
MAY 25 iS83
UNSOLICITED /;2 33 ~
DEPARTMENT OF AEROSPACE ENGINEERING AND ENGINEERING MECHANICS
UNIVERSITY OF TEXAS AT AUSTIN
AUSTIN, TEXAS 78712
ABSTRACT
~..-
~he effect of in-situ st~esses and layer properties \.I=.e-/
the propagation of a hydraulically induced fracture is studied
numerically in this paper. By comparing the maa~itude of
stress intensity factor at the fracture tips, our calculations
sho\v that, for a same ratio, the in-situ stress difference has a . -
greater effect to the movement of a fracture than the layer . '::.' ~; .'.
rna ter ia 1 difference. In adcition, our analysis indicates that
the in-situ stress containment effect on a hydraulic fracture
depends upon: (1) the ratio of the zone thickness to the
fracture he ight; (2) the rela ti ve magni tude of i n-s i tu stresses
among the reservoir zone, the high stress zone, and the zone
above it; and (3) the relative distance between the fracture tip
and the zone interface.
i
/2332-
INTRODUCTION
Warpinski, Schmidt <:1nd Northrop [1] * published an
interesting paper regarding the predominant factors influencing
the containment of a hydraul ic fracture. By examining the data
. obtained from hydraulic fracturing experiments conducted in a
mine tunnel complex, these authors concluded that: (1) a
material property difference between the reservoir rock and the
bounding formation is not sufficient to contain the fracture; and
(2) the magnitude and the gradient of the minimum in-situ
stress is the dominant factor affecting the movement of a
hydraulically induced fracture.
In fracture mechanics, the stress intensity factor at the
fracture tip is often used as an indicator of the stabi 1 i ty of
the fracture. The fracture tends to move towards the direction
that has a maximum stress intensity factor. Thus, by' comparing
the relative magnitudes of stress intensity factors along the
periphery of a crack, the up or downward motion of fracture in
layered rock medium can be determined. Using this idea, the
propagation of a hydraulically induced fracture in layered rock
medium was studied in detail by Yew and Lodde [2], and Lu and
Yew [3]. In these studies, the stress intensi ty factor at the
fracture tips due to the frac-fluid pressure was calculated by
taking the layer material differences into consideration. Using
the relative magnitude of stress intensity factor at the crack
tips to study the stability of a crack in layer materials is
.,
* Numbers in the bracket designate reference at the end of paper.
1 )2..-33'2-
LIST OF SYMBOLS
E: Elastic Modulus of Rock
H: Zone or Layer Thickness
K: Normalized Stress Intensity Factor
L: Half of Crack Length
P: Frac-fluid Pressure
V: Poisson's Ratio
0-: In-situ Stress
i i
) 233.;l
not new. The problem has been extensively studied by Erdogan et
al [4,5,6], Ashbaugh [7] and Goldstein and Vainshebaum [8] in
their investigations of cracks in layer composite materials. In
their studies, the major contributors to the stress intensi ty
factor at crack tips were: (1) difference between the layer
material properties, and (2) the size of the fracture and its
relati ve posi ton to the layer interface. Due to the nature of
the problems, the applied stress was usually assumed to be
uniform by these authors. The effect of rock layer property
differences on a hydraulically induced fracture was noted by many'
author s. Papers wr i tten by Daneshy [9,10], Hanson et· al [1-1,12] I
Abe et al [13], Simonson et al [14], and Cleary [15] pointed
out the importance of this effect. At the same time, the
importance of in-situ rock stress to the propagation of a
hydraulically induced fracture was also recognized and studied by
many authors. For this study, we cite works by Simonson [14],
Secor and Pollar [16], Pollard and Muller [17], and Lu and
Yew [3].
warpinski's experimental results [1] have singled out,
however, the in-situ rock stress as the predominant factor on
hydraulic fracture containment. Can this conclusion be taken as
a general statement in hydraulic fracturing? Is this containment
effect due to the in-situ rock stress or due to the difference
in layer properties influenced by the inherent zone thickness
and by the ratio of stresses o~ moduli between zones? This paper
addresses these questions. The finite element method was used to
perform the needed calculations for the stress intensi ty factor
at the fracture tips. As will be enumerated in later sections,
2 /2332-
our results partially support Warpinski's finding. In addition,
our results indicate that the in-situ stress containment effect
on a hydraulic fracture depends upon: (1) the ratio of the zone
thickness and the fracture height; (2) the rela ti ve magni tude
of in-si tu stresses among the reservoir zone, the high stress
zone, and the zone above it; and (3) the relative distance
between the fracture tip and the zone interface.
COMPUTATIONAL ARRANGEMENT AND PROCEDURE
Due to the geometrical complexity of the problem, an
analytical
proved to
approach using complex potential functions [3,4]
be inconvenient. The finite element method is,
therefore, chosen for analyzing this problem. In the forthcoming
analysis of the problem, the plane of the fracture was assumed to
be in a condition of plane strain. In view of the order of
magnitude differences between the fracture length, the height,
and the width of a hydraulically induced fracture, this
assumption appears to be an acceptable one except, perhaps, at
the very early stage of fracturing.
A finite element code TEXGAP [18), which performs
linearly elastic plane analysis, was used to evaluate the stress
intensity factor at the fracture tips. The code handles the
stress distribution near the fracture using a singular element
with built-in I/ff singularity. It is also well known that the
nature of singularity near the fr2ct'1re tip changes as the tip
approaches to the interface of layers or zones [4,5]. For this
reason, the fracture in this analysis was placed in a position
with its upper tip at a position a quarter of the total fracture
3 /2332--
length (0.5L) from the layer interface. Based on Erdogan [4,5]
and our own analysis [3] for a fracture at this position, the
stress intensity factor at the fracture tip is significantly
affected by the material properties on both sides of the
interface, but it keeps its square-root singularity. The
variation of stress intensity factor as the fracture tip
approaches the interface was stud ied in detai 1 by Erdogan [4].
It can be concluded that, based on his analysis, as the fracture
tip approaches the interface, the trend of stress intenisty
factor variation remains essentially the same in spite of the
fact that the degree of singularity at the tip is no longer a
square-root singulari ty. For this reason, we believe that our
results should provide a clear indication of how the stress
intensity factor at the fracture tips are affected by the
layers or the stressed zones above it.
A typical finite element grid used in the computation is
shown in Fig.1. The zone shown inside the dotted lines was the
zone where a repea ted
computation. Typically
started with a size
re-zoning process
the element near
of 0.25L x 0.25L ,
result was obtained when the element was
was applied during
the crack tip was
and a satisfactory
reduced to a size of
0.03l25L x 0.03l25L. In the TEXGAP code, the fracture surface
must be kept stress free. For this reason, a superposition
procedure was adopted. In the case of a homogeneous medium with
various in-situ stress zones, the superposition is straight
forward.
medium,
This is shown in Fig.2A. In the case of a layered
the situation is more complicated. In order to
4 /23sL
maintain continuity at the layer interface, the condition:
I - z),2 ~
I - zJ. ~ P2 Z ( I )
E, z: -2,
must be observed. A case of multi-layer superposition is shown
in Fig.2B. We further note here that, based on our numerical
experiment, the effect of finite edges on the stress intensity
factor becomes very small « 10.5%) when the size of domain is
larger than llOL where 2L is the fracture length.
RESULTS AND DISCUSSION
We first compute the stress intensity factors of a
fracture located near the interface of a two bonded half planes
as shown in Fig.2B. The computed normalized stress intensity
factor at tips A and B are KA = 1.10773;
Comparing with that obtained \lith the analytical method [3, 4]
KA = 1.10495, KB =1.10101010, the discrepancy is 2.65%.
The variations of the stress intensity factor of a
fracture located in a homogeneous medium under the action of
three representative cases of in-situ stress conditions are shown
in Fig.3, 4, and 5. Figure 3 shows a zone of high compressive
stress above the tip A. We first observe that the normalized
stress intensity factor at tip B has a value of approximately
one unit. The existence of a high stress zone above the fracture
decreases the stress intensity factor at tip Ai the fracture thus
te~d3 to migrate downwards because the stresS intensity factor
at tip B is larger than that at tip A. The fracture, in this
case, is contained from the upward motion by a high stress zone
above it. The containing effect of this high stress zone to
5
/2332-
the upward movement of fracture, however, decreases as the zone
thickness decreases. As expected, when the zone thickness reduces
to zero, the normalized stress intensity factor at tip A becomes
equal to that at tip B, i.e., KA = KB = 1. The fracture, in this
case, tends to expand as a circular fracture. The effect
of a high
A depends
stress zone on the stress intensity factor at tip
not only upon its zone thickness and its stress
magnitude, but also upon the relative magnitude of the in-situ
rock stress below and above this high stress zone. This effect
is portrayed in Figure 4. In this figure, the magnitude of
in-situ stress in the high stress zone is taken to be three
units. The variation of the stress intensity factor at tip A is
plotted as a function of the ratio of the zone thickness and
the half fracture height H/L, and the magnitude of in-situ
stress ~ above the zone. Figure 4 clearly shows that the stress
intensity factor at tip A increases as the ratio H/L and the
magnitude of the in-situ rock stress above the zone U
decreases. This implies that (when the in-situ rock stress above
the high stress zone is small in comparison with the reservoir
in-si tu stress, and the thickness of this high stress zone
is thin in comparison with the fracture height) the magnitude
of stress intensity factor at tip A can become larger than
that at tip B; and the fracture can thus move upward in spite of
a high stress zone directly above it.
Figure 5 shows an extreme case of low in-situ stress zone
sandwiched between the reservoir zone and a moderate high stress
zone. It is interesting to observe that the magnitude of
stress intensity factor at tip A is again dependent upon the
6
ratio of the zone thickness relative to the fracture height.
When this ratio is small, the stress intensity factor at tip
A is less than one; and the fracture can not move upward in
spite of a low stress zone above it.
The effect of relative moduli and thickness between
layers to the magnitude of stress intensity factor is shown in
Fig.6. It clearly shows that the stress intensity factor at tip
A depends upon the relative moduli between the layers and the
H/L ratio. A soft layer (i.e., a layer with a lower modulus)
directly above the fracture tends to make the stress intensity
factor at tip A higher than one unit; and- the fracture thus tends
to move upward. This upward strength increases as indicated
by a gradual increasing of the magnitude of stress intensity
factor when the layer thickness increases. This upward strength
can, however, be suppressed by a hard zone above it. Figure 6
shows that when the H/L ratio is small, the existence of a.
hard zone above the soft layer can make the stress intensity
factor at A less than one unit; and thus prevents the upward
motion of the fracture.
Figure 6 further shows that the stress intensity factor
at tip A decreases as the thickness of the above hard layer
increases. A hard layer above the fracture has, in general, a
containing effect on the upward motion of the fracture. This
containing effect is, however, affected by a soft layer above it.
When the layer thickness is not large, the existence of a soft
layer above it can make the stress intensity factor at tip A
larger than one unit which nullifies the containing effect of the
hard layer. This effect is clearly shown in Fig.6.
7
CONCLUSION
Our computational results indicate that the in-situ rock
stresses are indeed an important factor in the design of a
hydraulically induced fracture. The containment effect of in-
situ stresses to a fracture, according to our
dependent upon the following four factors: (1) the
analysis, is
distribution
of in-situ stresses in the neighborhood of fracture ranging from
three to four fracture height; (2) the ratio of the fracture
height to the thickness of the stress zone adjacent to the
fracture; (3) the relative magnitude of in-situ stresses between
the reservoir zone and in zones adjacent to it ; and (4) the
rela ti ve distance between the fracture tip and the boundary of
stress discontinuity.
The difference in layer properties appears to have the
same containing effect on a fracture as the in-situ stresses.
However, by comparing the magnitudes of stress intensity factors
resul ting from the difference in in-si tu stresses and from the
difference in layer properties, our results indicate that the in
situ stress has a greater effect on the magnitude of stress
intensity factor than the layer material properties. In other
words, for a same ratio, the stress intensity factors calculated
based on the in-situ stress differences have a larger value than
those calculated based on the layer material differences. The
in-situ stresses, therefore, appear to have a more dominant
influence on the movement of a hydraulically induced fracture
as suggested by Warpinski et al [1]. However, the agreement is
qualitative because the question of how the in-stresses and the
8 /233'2-
layer properties are relatec. remains unanswered. If one accepts
a simplified model that the in-situ stress is caused by the
tectonic plate movement and the so proc.ucec stress is supported
by the layered medium, then, under a plane strain condition,
the interfacial displacement continuity equation, Eq. (1),'
clearly shows that a harder layer will carry a higher in-situ
stress depending upon the moduli ratio between the layers.
More studies are needed in this area to unravel the precise
relationship betweeen the layer properties and the in-situ
stresses.
9
ACKNOWLEDGEMENT
This study was conducted pursuant to an agreement between
the University of Texas at Austin and Exxon Production Research
Company. The guidance given by Dr. D.E. Nierode of Exxon
Production Research Company is gratefully acknowledged.
10 )2332
REFERENCES
[1] N.R. Warpinski, R.A. Schmidt, and D.A. Northrop, "In Si-tu
[ 2 ] C.H.
Stresses: The Predominant Influence on Hydraulic F r act u r e Con t a i n men t • II ~ P E / DOE '§"2.l~ ~ 0 E1:. e !y 0 ! Petroleum Engineers, 1982, pp.83-94.
Yew and P. Loddle, "Propagation of a Hydraulically Induced Fracture in Layered Medium." submi tted to SPE for publication.
[3] C.K. Lu and C.H. Yew, "On Bonded Ha 1 f-planes Conta i n i ng Two Arbitrarily Oriented Cracks: A Study of Containment of the Hydraulically Induced Fractures. 1I submitted to SPE for publication.
[4] F.Erodogan and v.Biricikoglu, IITwo Bonded Half Planes with a Crack Going through the Interface. 1I International Journal of Engineering Science, Vol.ll, 1973, pp.745-766.
(5] F. Erodogan and o. Aksogan,"Bonded Half Planes Containing An Arbitrarily Oriented Crack.1I International Journal of Solids and Structures, Vol.10, 1974, pp.569-585.
(6] T.S. Cook and F. Erdogan, liS tresses in Bonded Materials with a Crack Perpendicular to the Interface." International Journal of Engineering Science, VOl.10, 1972, pp-677-697.
[7] N. Ashbaugh, "Stress Solution for a Crack at an Arbitrary Angle to an Interface." International Journal of Fracture, Vol.ll, No.2, Apri1.-T97S;-p.205.- ------
[8] R. V. Goldste in and V.M. Va inshelbaum, "Ax i symmetr ic Problem of a Crack at the Interface of Layers in a Multi-layered Medium." International Journal of Engineering Science, Vol.14, No.4, 1976, pp.335-342.
[9] A.A. Daneshy, "Hydraul ic Fracture Propag a t i on in Layered Formation." Society of Petroleum Engineers Journal, February 1978, pp.33-41.
[10] A.A. Daneshy, "Three-Dimensional Propagation of Hydraulic Fractures Extending from Open Holes." Application of Rock Mechanics, Proceedings of ASCE 15 th Symposi urn on Rock Mechanics, ed. by Haskin, pp.157-179.
[11] M.E, Hanson, G.D. Anderson, R.J. Shaffer, D.O. Emerson, H. C. Heard, and B.C. Haimson, "Theoretical and Experimental Research on Hydraulic Fracturing. 1I
Proceedings, Fourth Annual DOE Symposium on Enhanced Oil and Gas Recovery and Improved Drilling Method, Aug. 1978, Tulsa, OK.
11
[ 12 ] M.E. Hanson, R.L. Shaffer, Var ious Parameters on Society I of Petroleum pp.435-443.
amd G.D. Anderson, "Effects of Hydraul ic Fractur ing Geometry." Engineers Journal, Aug. 1981,
[13] H. Abe, T. Mura, and L.M. Keer, "Growth Rate of a PenneyShaped Crack in Hydraul ic Fractur i ng of Rocks." Journal of Geophysical Research, Vol.81, No.29, Oct. 1976, pp.5335-5340.
[14] E.R. Simonson, A.S. Shou-Sayes and R.J. Clifton, "Containment of Massive Hydraulic Fractures. " Society of Petroleum Engineers Journal, Feb. 1978, pp.27-32.
[15] M.P. Cleary, "Primary Factors Governing Hydraulic Fractures in Heterogeneous Stratified Porous Formation." ASME paper no. 78-Pet-4 7, Paper presented at the Energy Technology Conference and Exhibition, Houston, Texas, 1978.
[16] D.T. Secor, Jr., and D.O. Pollard, "On the Stability of Open Hydraulic Fractures in the Earth's Crust." Geophysical Research Letters, Vol.2, No.ll, Nov. 1975, pp.510-5l3.
[17] 0 • .0. Poll ard, and O.H. Muller, "The Effect of Grad ients in Regional Stress and Magma Pressure on the Form of Sheet Intrusions in Cross-section." Journal of Geophysical Research, Vol.81, No.5, Feb. 1976, pp.975-984.
[18] R.S. Dunham, and E.B. Becker, "The Texas Grain Analysis Program." TICOM Report 73-1, The University of Texas at Austin, August 1973.
12 /'b332-
LIST OF FIGURES
Figure 1. Finite Element Grid
Figure 2. An Illustration of Superposition of Fluid Pressure and In-situ Stress
Figure 3. The Effect of a High In-situ Stress Zone Above the Fracture
Figure 4. The Effect of Relative In-situ Stresses Below and Above the High Stress Zone
Figure 5. The Effect of a Low In-situ Stress Zone Above the Fracture
Figure 6. The Effect of Layers
13 J 2332-
OL
..
A
T '- 2L
1 ~ crack
eleme "/" rezone region
,.
Fi g. I Fini fe Element Grid.
40"
T H 3cr
...L O.5L
A
T cr -
2L -
B 1 (A) Superposition of pressure
with in - situ stress.
H E2
««<t«« O.5L E
A -r ,
B
I 2L I
1
»»»»»
«««
(B) Superposition of pressure In layer medium.
4cr -p
3cr - p
p
+ p-O"
»»»))
10--_ E'3 I - 71,2 ......--- 2 P 1--- E, I-V:;
«««( E2 I_V,2 - P EI I-Vt
+
Fig. 2 An illustration of superposition of fluid pressure
and in - situ stress.
} 2352....
C\I
.. co cr' 00 tU « L&..
>-~ 1-' _0 en z LLJ t-Z -q-
o
C\I
o
0 0.4
Fig.3
0--0- = 6 p A-(j = 4 P x--(j = 3p
O.S
The effect of the fracture.
a
p
* H (j
A + O.5L
2P~t p
B
1.2 1.6 2.0 2.4 2.8 H/L
high in-situ stress zone above
/2 332--
« ~
a::: 0 l-e...> « LL.
)0-
l-en z w r-z
v N
0 N
<D -
N . -
00 0
qo
0--0"' = b,--(j = C--(j = X --Ci =
.25p
.5 p , I p H 3p
A
~ 1.5p
2p 2L
1 p
B
o+-------~------~------~-------,-------,~----~
0.4 0.8 1.2 1.6 H/L
2.6 2.4 2.8
Fig.4 The effect of relative in- situ stresses below and
above the high stress lone.
}2'3sZ
a:: o to-~o LL....:
>-t-en Z 1JJ(1)
~O
ex)
o
~
°0.4
Fig. 5. The the
H
2p
A~ !~L 8
0.8 1.2
effect of a low fracture.
1.5p
(j =0 -- P -. -
1.6 2.0 2.4 2.8 H/L in- situ stress zone above
I 233'2..
0 -
(t)
0 -
a:; o t-U <tv I.t..O - -)om
t-en Z WC\I ""'0 z...;
o Q
(t)
o --E = 3
6,-- E :: '3
c--E = 3
E 3 t tT t H E
2,1J
A lO.5 L
EI ' E2 =E,/20
~lL E 1/ 2.0, E2 :: 2E, p Elt V'
2E, ' E 2= E,/20 8
m+-______ ~ __ ----~------~------~----__ ~----__ °0 0.4 0.8 1.2
H/L
Fig.6 The effect of layers.
1.6 2.0 2.4