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