Hydraulic Fracture Geometry: Fracture Containment in Layered Formations H. A. M. van Eekelen, Koninklijke/Shell Exploratie en Produktie Laboratorium

Abstract One of the main problems in hydraulic fracturing technology is the prediction of fracture height. In par­ticular, the question of what constitutes a barrier to ver­tical fracture propagation is crucial to the success of field operations.

An analysis of hydraulic fracture containment effects has been performed. The main conclusion is that in most cases the fracture will penetrate into the layers adjoining the pay zone, the depth of penetration being determined by the differences in stiffness and in horizontal in-situ stress between the pay zone and the adjoining layers. For the case of a stiffness contrast, an estimate of the penetration depth is given.

Introduction Current design procedures for hydraulic fracturing of oil and gas reservoirs are based predominantly on the frac­turing theories of Perkins and Kern, 1 Nordgren,2 and Geertsma and de Klerk. 3 In the model proposed by Perkins and Kern, and improved by Nordgren, the for­mation stiffness is concentrated in vertical planes perpendicular to the direction of fracture propagation. The fracture cross section in these planes is assumed elliptical, and the stiffness of the formation in the horizontal plane is neglected. In the model proposed by Geertsma and de Klerk, the stiffness of the formation is concentrated in the horizontal plane. The fracture cross section in the vertical plane is assumed rectangular, and the stiffness in the vertical plane is neglected. In both models, the fluid pressure is assumed a function of the distance from the borehole, independent of the transverse coordinates. The theory by Perkins and Kern is more appropriate 4 for long fractures (LlH~ 1, where Land H are length and height of the fracture), whereas the model by Geertsma and de Klerk is applicable 4 for short fractures, LI H::; 1.

0197·752018210006·9261 $00.25 Copyright 1982 Society of Petroleum Engineers of AI ME

JUNE 1982

The main shortcoming of these fracture-design pro­cedures is that they assume a constant, preassigned frac­ture height, H. The value of H has a strong influence on the results for fracture length, fracture width, and prop­pant transport. Usually, the estimated fracture height is based on assumed "barrier action" of rock layers above and below the pay zone. This situation is rather un­satisfactory. Moreover, if these layers do not contain the fracture, large volumes of fracturing fluid may be lost in fracturing unproductive strata, and communication with unwanted formations may be opened up.

Whether an adjacent formation will act as a fracture barrier may depend on a number of factors: differences in in-situ stress, elastic properties, fracture toughness, ductility, and permeability; and the bonding at the inter­face. We analyze these factors with respect to their relative influence on fracture containment.

Differences in in-situ stress and differences in elastic properties affect the global or overall stress field around the fracture, and, hence, the three-dimensional shape of the fracture. This shape, together with the horizontal and vertical fracture propagation rates, determines the fluid pressure distribution in the fracture, which in tum affects the stress field around the fracture. Consequently, the elastic stress field, the fluid pressure field, and the frac­ture propagation pattern are intimately coupled, which makes the fracture propagation problem a complicated one.

Whether at a certain point of the fracture edge the frac­ture will propagate is determined by the intensity of the stress concentration at that point. This stress concentra­tion depends on the global stress distribution in and around the fracture, but it also is affected directly by local ductility, permeability, and elastic modulus in the tip region.

For example, idealized linear elastic fracture theory predicts that the stress-intensity factor approaches zero as a fracture approaches an interface with a layer of

341

higher stiffness (higher shear modulus) 5-7 ; this effect is independent of the fluid pressure distribution in the frac­ture. Similarly, the stress intensity factor effectively decreases when the fracture crosses into a layer of higher ductility 8 or lower permeability. 9 Another class of local effects results from the nature of the interface between two layers. For example, if the layers are poorly bonded, slip may occur, leading to blunting of the crack tip and subsequent crack arrest. The effect of a "smeared-out" interface, where properties change gradually from one layer to the next, may differ from that of a more abrupt transition.

It is shown in the next section that it is unlikely that the local effects mentioned are of primary importance in determining the geometry of hydraulic fractures. It is more probable that in-situ stress and stiffness differences between layers, through their influence on fracture cross section and fluid pressure distribution, are the main fac­tors determining fracture shape.

Stress Intensity Factor In the linear elastic theory of fracture, the stresses around the tip of a crack are singular with r- 1/2

, where r is the distance to the crack tip. The strength of the singularity is measured by means of the stress intensity factor, K, which, for a tensile (Mode 1) crack, is defined as

where a l' is the tensile stress on the crack axis ahead of the tip.' The value of the stress intensity factor, K, depends on the fracture geometry and on the load ap­plied. For instance, for an infinitely long crack of height H, internally pressurized by a fluid and propagating through a homogeneous material: 10

K=1.256pJH, .......................... (1)

while, for a coin-shaped crack of radius H12,

K=0.80 6pJH.

In these formulae, 6p =p - S h' where p is the fluid pressure inside the crack and S h is the formation stress normal to the plane of fracture.

The fracture propagates if the stress intensity factor K reaches a critical value K c' which is assumed to be a material property called critical stress intensity factor, fracture toughness, or fracturability. Measured values of KoinMPa·m ll

, (psi-in.II'),are

1.04<Kc<1.81, (950 < K c < 1,650) for siltstone, 11.12

0.44<Kc< 1.76, (400<Kc < 1,600) for sandstone, 11-13

0.44<Kc < 1.04, (400 < K c < 950) for limestone, 14-17

0.33 < Kc < 1.32, and (300 < K c < 1,200) for shale. 11.12,15,18

All these values have been measured under low con­fining pressures; under downhole conditions, the frac­ture toughness may be somewhat higher-e.g., 14 a fac­tor of 1.6 at 24 MPa (3,500 psi). However, the main

342

point is that the range of measured values of fracture toughness is very narrow-all but one of the values of Kc in Refs. 11 through 18 are within a factor of two from 0.88 MPa' m liz (800 psi-in. II,).

As mentioned in the Introduction, existing theories of hydraulic fracturing assume a constant fracture height, H. Let us consider the case where the material is homogeneous, the fracture length L is much larger than H (as is the case in most fracturing designs), and the fluid pressure inside the crack is a function p(x) = S h + 6p(x) of the distance x to the borehole. The stress intensity factor at the upper and lower edges of the crack may then be approximated by

K(x) = 1.25 6p(x)JH,

while the stress intensity factor at the crack front (x=L) will be slightly larger than that of a coin-shaped crack of radius HI2 and may be approximated by

where 6p L is some average value of 6p(x) over the region near the crack front.

If the fluid overpressure, 6p, were constant over the crack (independent of x), it would follow that the stress intensity factor at the upper and lower edges of the crack would be about 25 % higher than at the crack front. [Cleary 19 gives the ratio of stress intensity factors as K(x)IK(L)=.J(2L1H), which may be much larger than 1.25. His analysis is based on an elliptical cross section in the x-z plane. It is more likely that a contained long crack would have an approximately semicircular end.] Consequently, the crack would start propagating in the vertical direction, unless adjoining layers contain the fracture by decreasing the value of K or increasing the value of K c' The width of the fracture would be given by

I-v I-v W=--H6p:=: --Kc.JH,

G G where G is the shear modulus and v is Poisson's ratio for the formation. Taking a representative case with

Kc 1 MPa' m \/, (910 psi-in. II,), G 104 MPa (1.45 x 106 psi), v = 0.25, and

H 50 m (164 ft),

we obtain a fracture width W=0.50 mm (0.02 in.), at a fluid overpressure 6p :=: 0.14 MPa (21 psi).

Obviously, a fracture 0.50 mm (0.02 in.) wide would not admit proppant. To increase fracture width, use is made of highly viscous fracturing fluids and high pump­ing rates. In this way, a viscous pressure drop is created along the fracture, which forces the crack to open wider. Again taking a representative case, with fluid viscosity 1'/ =0.1 Pa' s (100 cp) (a hundred times the viscosity of water), pump rate Q=0.04 m 3 /s (15 bbllmin), and frac­ture length L=200 m (56 ft), Nordgren's theory without fluid loss gives, for a single-wing fracture, the fracture width at the wellbore

SOCIETY OF PETROLEUM ENGINEERS JOURNAL

(I-v )

W=2.75 aY/QL '4 =7.7 mm (0.30 in.)

and, for the overpressure at the wellbore,

G W 6p=----=2.0 MPa (296 psi).

I-v H

However, with this value of 6p, the stress intensity factor at the upper and lower edges of the fracture would be

K = 18 MPa . m '/2 (16,400 psi-in. '/2),

which is an order of magnitude larger than the critical value, Kc. Of course, it is impossible for K to exceed K c' and what really happens is that the crack edge con­tinually keeps ahead of the fluid, so that the over­pressure, 6p, is not applied right up to the crack edge (in fact, in some laboratory tests 20 the fluid was found to occupy only 60 to 70% of fracture length). It is clear that differences in fracture toughness, K c, between layers will not contain such a fracture and that local effects that decrease the stress intensity factor, K, (ductility, permeability, stiffness contrast) will not reverse this situation; whenever K drops below Kc, the fluid catches up with the crack edge until K is again equal to Kc and the fracture edge starts moving again.

Thus, the problem of fracture containment in hydraulic fracturing operations may be summarized as follows. To obtain reasonable fracture widths that allow proppant to enter the fracture, use is made of highly viscous fluids and high pumping rates. The high pressures involved cause the stress intensity factor at the upper and lower edges of the fracture to be (potentially) an order of magnitude higher than the critical value, K c. Local ef­fects around the fracture edge, which decrease K or in­crease K c' will not be sufficient to contain such fractures within the pay zone. Apart from interface slippage, which is expected to occur only at shallow depths (discussed later), the only effects that might (partially) contain such fractures are those that influence the overall stress and deformation patterns around the fracture. These global effects are differences between adjoining layers as regards (1) elastic stiffness and (2) in-situ stress.

Local Containment Effects Around the Crack Edge In this section, a more detailed discussion is given of the local effects that may change the (effective) stress inten­sity factor as a crack approaches or crosses an interface between two layers.

Ductility

According to the linear elastic theory of fracture, both stresses and strains around a fracture tip are singular with r - '/2, where r is the distance to the crack tip. Singular strains correspond to a dislocation or discontinuity, but singular stresses are physically impossible. To eliminate the stress singularity, one can assume a plastic zone 8

JUNE 1982

around the crack tip, * where the stresses are limited by a "yield condition."

A discussion of the role of plasticity in fracture mechanics has been given by Rice 8 for the case of a Mode 3 crack (out-of-plane shearing). The material is assumed linearly elastic up to a certain yield value of the shear stress, T (), or the shear strain, 'Y (). After yield, the shear strain increases at constant shear stress; this is what happens in the plastic zone around Rice's crack tip. It is assumed further that the fracture propagates if the strain at a fixed microstructural distance, p s, ahead of the crack tip reaches a value 'Y j= (1 + Dh ()' where D is the ductility (by definition). It can be shown 8 that the resistance to fracture propagation then increases with crack length until an asymptotic value is reached; the in­crease is larger for larger ductility, D. In this way, a duc­tile material such as shale may, in principle, stop a run­ning crack. In essence, the effect is caused by energy dissipation in the plastic zone around the crack tip. It may be interpreted as a gradual increase of the (effective) fracture toughness, K c' for a running crack.

The applicability of Rice's analysis to a Mode 1 (hydraulic fracturing) crack is uncertain, but it is clear that some plastic energy dissipation will occur. However, the question is whether this effect has been in­cluded already in the measured values of fracture toughness, K c. The asymptotic fracture propagation pressure is reached when the crack length is large com­pared with the microstructural distance, p s . The microstructural distance probably will be of the order of millimeters or less, whereas the characteristic crack length in fracture toughness tests is of the order of cen­timeters. II Hence, it seems likely that in these tests asymptotic fracture propagation is approached and that the measured value of fracture toughness would not in­crease substantially if the sample size were increased -e.g., to 1 m (3.3 ft). Although it still would be useful to investigate the dependence on sample size of the measured value of Kc in jacketed burst tests, 11 we con­clude that the effect of ductility on fracture propagation already has been accounted for in the measured value of fracture toughness and that it does not make a further contribution to fracture containment.

Permeability

The simplest (linearly elastic) description of the stress/strain behavior of fluid-saturated porous rock is by means of poroelasticity relations formulated by Biot 23 ; a recent review has been given by Rice and Cleary. 24

There are two limiting cases in which poroelastic response corresponds to the response of a classical elastic solid. For slow deformation (slow in comparison with the time scale for diffusive transport of pore fluid), the material deforms without changes in pore pressure and behaves as an elastic solid with shear modulus, G, and Poisson's ratio, Vd (d for drained). In the limit of very rapid deformation, there is no time for fluid flow,

* An alternative is to introduce cohesive forces in a small region behind the crack tip,21 the critical stress intenSity factor then is replaced by a modulus of cohesion. In the case of hydraulic fracturing, a simplified version of the cohesive force theory is ob­tained by assuming that the fracturing"fluid (or the fracturing fluid pressure) does not extend all the way out to the crack tip. 2 The compressive in-situ stresses then act as cohesive forces around the crack tip. This concept has been used by Geertsma and de Klerk in their model of hydraulic fracturing. It deviates from the original cohesive force theory in that the modulus of cohesion (or fracture toughness) of the material itself is effectively equal to zero.

343

and the material behaves as a massive impenneable solid with shear modulus, G, and Poisson's ratio I'u>I'd' which means that volumetric stiffness is larger for un­drained than for drained defonnation; the shear stiffness is the same. To give an idea of the magnitude of the ef­fect, we quote 24 three pairs of values for I' u and I'd:

Ruhrsandstone: I'd=0.12, I'u=0.31 Berea sandstone: I'd =0.20, I' u =0.33 Weber sandstone: I'd=0.15, I'u=0.29 Rice and Simons 9 have solved the problem of a plane

strain shear (Mode 2) crack, which propagates at con­stant speed through a homogeneous poroelastic material; a short and lucid discussion of their results has been given by Rice. 25 It is found that the propagation pressure, 6p, is a function of vale, where v is the crack speed, a is the crack length, and c is the poroelastic dif­fusivity.24 The precise shape of the function depends on the assumed size of the process zone around the crack tip. For our purpose, it is sufficient to note that the max­imum increase in propagation pressure caused by in­creasing crack speed is bounded as follows.

With the values of Poisson's ratio given in the preceding paragraph, the effect of crack speed on propagation pressure is less than 30%.

For a Mode 1 (hydraulic fracturing) crack, the effect would be expected to be somewhat larger than for a Mode 2 crack because the volume stiffness plays a more important part in tensile cracking than in shear cracking. Nevertheless, analysis of the problem 26 has shown that the upper bound of the above inequality is also ap­plicable to Mode 1 cracks. Hence, the difference be­tween fast and slow (undrained and drained) propagation pressures, 6p, should be expected to be less than 30%. Moreover, part, if not all, of the effect already is ac­counted for in the measured value of fracture toughness, Kc. This applies especially for shales, for which the dif­fusivity is very low because of low penneability, and the value of vale in the fracture toughness test will be large. Consequently, we conclude that a penneability contrast does not give a significant contribution to fracture con­tainment beyond that which is already contained in the values of K c .

Stiffness Contrast

According to idealized linearly elastic fracture theory, the stress intensity factor at the tip of a crack approaches zero if the crack approaches an interface with a stiffer material; an analytical derivation of this effect has been given by Hilton and Sih.5 In fact, if the crack tip is located at the interface, the stress singularity is no longer of the fonn r-'/2. For example, 6,7 ifv2 =1' 1 =0.3 and the stiffness ratio G2 /G 1 is equal to three, the stresses are singular, with r-O.4. It has been stated that this means that a crack would never cross an interface with a stiffer material, 27 but it is obvious that such a general statement cannot be correct.

If the stress intensity factor approaches zero when the crack tip approaches an (infinitely sharp) interface with a

344

stiffer material, it is doubtful whether it is correct to use a fracture propagation criterion K = K c that has been derived for homogeneous materials. Furthennore, the finite size of the process zone at the crack tip, the finite size of the transition zone between the layers, and the ex­istence of natural flaws in the material all should be taken into account. Finally, both laboratory 20 and field 28,29 tests indicate that fractures do break through into layers with higher shear modulus. This conclusion has been confinned in fracturing tests by the author on layered blocks of gelatin, in which a stiffness contrast up to a factor of four did not act as a fracture barrier. In these and other 20,30 laboratory tests on layered systems, crack arrest at interfaces was caused by slippage due to poor bonding, rather than by stiffness contrast. It may be concluded 28,29,31,32 that a stiffness contrast, by itself, does not constitute a barrier to propagation of fractures that are driven by viscous fluids at high pump rates.

As noted in the Introduction, however, a stiffness con­trast between adjacent layers also has an influence on the overall stress field in and around the fracture and on the fracture width. This subject is discussed in the next sec­tion, together with the effect of in-situ stress differences.

Interface Slippage

As mentioned earlier, crack arrest at interfaces in laboratory tests on layered systems usually is caused by interface slippage. This phenomenon can be prevented by increasing the compressive stress nonnal to the inter­face, which improves the frictional resistance to slip­page. For example, in tests on layered samples of Nug­get sandstone,30 a crack crossed both a rough and a smooth interface under a nonnalload of 6.9 MPa (l,000 psi), it crossed only the rough interface at a nonnalload of 4.3 MPa (625 psi), and it crossed neither the rough nor the smooth interface at a nonnal load of 3.4 MPa (500 psi). No bonding agent was used in these tests, so the interface had no tensile strength.

In field operations, one may expect crack arrest by in­terface slippage to occur at shallow depths if the layers are separated by a sharp interface. At greater depth, however, high frictional resistance caused by the over­burden load probably will prevent interface slippage, so that this crack arrest mechanism will not be operative. The depth up to which interface slippage can occur will be dependent, to some extent, on rock properties.

Global Containment Effects In this section, we discuss global containment effects caused by differences between adjoining layers as regards elastic stiffness and in-situ stress. These dif­ferences do not keep a fracture from crossing an interface between two layers, but may restrict the depth of penetration into the adjoining layer.

Difference in Elastic Stiffness

Let us consider a fracture of constant height, H, and in­creasing length, L, that propagates from a wellbore into a homogeneous fonnation with shear modulus, G, and Poisson's ratio, 1'. The fluid pressure in the wellbore~ p w, is kept constant, so that the flow rate, Q, varies with time; fluid loss into the sides of the fracture is neglected.

For a long fracture (L> H), the fracture width at the wellbore is constant and given by 1,4

SOCIETY OF PETROLEUM ENGINEERS JOURNAL

W=H(1-v) 6.p/G, ........................ (2)

where 6.P=Pw-Sh, with Sh the fonnation stress perpendicular to the plane of fracture. Assuming that the fluid pressure at the end of the fracture is equal to S h , the pressure drop, 6.p, (for laminar, Newtonian flow) may be written according to Nordgren 2,4:

_ [( G ) 3 4J ';'\ 6.p-2.75 I-v YJQLlH , ............ (3)

where YJ is the fluid viscosity, and Q is the flow rate into (one wing of) the fracture. From Eq. 3, we find the flow rate, Q(t), as a function of fracture length, L(t). On the other hand, the flow rate is equal to the increase in frac­ture volume with time 2,4.

d

h

I --- L-

Q= -(0.59 WHL) Fig. 1-Hydraulic fracture in a three-layer system. dt

2 6.p dL =0.59H (1-v)-- . .................. (4)

G dt

Substituting Q from Eq. 3, we obtain the fracture­propagation rate

dL H2 (6.p) 3

-=0.30-(1-v)2 __ . ............... (5) dt L YJG2

Fora shortfracture (L<H), a similar derivation with the theory by Geertsma and de Klerk 3,4 gives

dL =0.190L(1-v)2 (6.p) 3 ................. (6)

~ YJG 2

The two expressions for dLidt agree for an almost square (two-winged) fracture with 2L1H=0.8.

Comparing Eqs. 5 and 6, we see that for constant 6.p, the propagation rate for a short fracture increases with time or fracture length, whereas the propagation rate for a long fracture decreases with time or fracture length. For both short and long fractures, the propagation rate is inversely proportional to G2 . Other things being equal, this means that a fracture propagates four times faster in a layer with modulus G, than in a layer with modulus G2 =2G,.

However, in most cases, other things will not be equal. To see this, consider a vertical fracture in a fonna­tion consisting of a central layer of height h and modulus G" sandwiched between two layers with modulus G 2 > G, (see Fig. 1). If the fluid overpressure, 6.p, is independent of the vertical coordinate, z, a reasonable assumption for the maximum width of the fracture seems to be (see Eq. 2)

h 6.h h' W=(1-v)6.p(-+--)=(1-v)6.p-, .. (7)

G, G 2 G,

where the effective fracture height, h', is equal to h(1 +G, 6.h/G 2h). Note that W has the correct limits for G 2 = G, and G 2 > > G I. Concerning the vertical cross section of the fracture, we make two further assumptions (see Fig. 2):

I. In the central layer, the fracture is part of an ellipse with major axis, h', and minor axis, W.

JUNE 19X2

h w

Fig. 2-Fracture cross section in a three-layer system.

h'

H h

20

Fig. 3-Hydraulic fracture growth in a three-layer system-fracture length vs. fracture height.

2L II

2. In the outer layers, the fracture is part of an ellipse with major axis h+ 6h and minor axis given by the re­quirement of continuity at the interfaces. One then finds that the shape of the fracture in the outer (G 2 ) layers is the same as if the material were homogeneous with shear modulus

( I + 6 hl2h ) [/,

G'= G, G2 ............. (8) l+G,6h12G2h

For small 6hlh, the crack width at the interface is a fac­tor .,) G, I G2 smaller than it would be in a homogeneous material with modulus G" whereas it is a factor .,) G2 I G, larger than it would be in a homogeneous material with modulus G2 .

These considerations are used to obtain an estimate for the horizontal and vertical propagation rates of the frac­ture of total length 2L and height H=h+ 6h depicted in Fig. I. We assume that 6 h < h and that the outer layers are appreciably stiffer than the central layer: G2 > > G, . In that case, the crack in the outer layers will be of small width and volume compared with the crack in the central layer. In addition, more than 80% of the pressure drop in the horizontal direction occurs in the region x> Ll2, in which fracture penetration of the outer layers will be small or zero. Consequently, the horizontal crack prop­agation rate, dUdt, will be influenced hardly at all by the crack in the outer layers, and will be given by Eq. 5 with G=G" h instead of H, and 6p the overpressure in the wellbore.

Regarding fracture propagation in the vertical direc­tion in the region around the wellbore, note that almost all the pressure drop in this direction occurs in the outer layers where the crack is narrow and has the same shape as in a homogeneous material with modulus G'. For small 6 hl2h, we have G' =.,) G, I G2 constant, and we may assume that the vertical propagation rate is given approximately by Eq. 6, with L replaced by Hand G2 by G,2 =G, G2. In that case, the ratio of the horizontal and vertical propagation rates is given by

dL 3 h 2 G 2 =--- ......................... (9)

dH 19 LH G,

which is independent of 6p, and proportional to G 2 /G, instead of (G2 I G, ) 2. Integration of Eq. 9 with the initial condition 2L=h at H=h gives

346

( 24 G2 H) [/,

2L=h I+19~logh ................. (10)

A somewhat more accurate calculation, where we use all of Eq. 8 for G', modify Eq. 6 to account for the fact that G' changes with H, and neglect terms of order (G,6hI2G2h)2 compared with terms of Order I, gives as the final result,

( 12 G2 ) [/,

2L=h I+---F , ................... (II) 19 G,

with

F=(I+~~ )Iog H 2 G2 h

+~(H _I)_~~(H2 -I). 4 h 8 G2 h 2

A plot of Hlh against 2Uh is given in Fig. 3 for G2 /G, =5, 10, and 20. After the fracture has reached the interface (2L=H=h), it initially grows mainly in the horizontal direction. However, the more elongated the fracture becomes, the more favorable the conditions for propagation in the vertical direction. As a consequence, the elongation, 2LIH, reaches a maximum value after which it slowly declines. This is illustrated in Fig. 4, where the elongation has been plotted as a function of fracture length. A plot of the maximum elongation (2UH)max as a function of G2 /G, is given in Fig. 5 (solid curve); for G2 I G, > 5 the maximum elongation is approximately equal to V2.,)G 2 /G, (dashed curve). It is seen that, to obtain a fracture with elongation 2LIH=3, one would need a modulus contrast, G2 /G, >40.

Although these results should be considered order-of­magnitude estimates (the expressions for horizontal and vertical propagation rates are only very approximate), they do indicate that the containing effect caused by an elasticity contrast is not as strong as one might hope. Although in principle the propagation rate of a fracture at given overpressure 6p is proportional to C-2 , the actual effect in the geometry of the layered system is far smaller. One reason for this is that the crack width in the outer layers is estimated to be a factor .,) G2 I G, I a rg e r than it would be without the more compliant central layer. A second reason is that the crack in the outer layers is relatively long in the horizontal direction, so that transverse elasticity does not play a role and the crack becomes wider as it goes deeper. Consequently vertical fracture growth at constant 6p is an accelerating process (dHldt - H), whereas horizontal fracture growth is a decelerating process (dUdt - II L).

We conclude that a stiffness contrast between the layer to be fractured and the adjoining layers will cause the fracture to assume a horizontally elongated shape. For reasonable stiffness ratios, however, the elongation ratio probably will not be larger than two or three.

So far it has been assumed tacitly that Poisson' s rati~ in the bounding layers, 1'2, is the same as in the central layer, 1', .

If both shear modulus G and Poisson's ratio I' vary be­tween the layers, the assumption of Eq. 7 about the

SOCIETY OF PETROLEUM ENGINEERS JOURNAL

,

0

.,

0

" ..-

/ --- --:-;=20

/ I ----

--~"'O G,

/--~~:~

Fig. 4-Hydraulic fracture growth in a three-layer system-fracture elongation vs. fracture length.

width of the fracture may be replaced by

" ,

The analysis then remains unchanged, provided that parameter G is replaced everywhere by r=G/(1-v). Consequently, the conclusion that the containing effect caused by a stiffness contrast is relatively weak also ap­plies to a contrast in Poisson's ratio.

Differences in In-Situ Stress

The effect on fracture geometry of a contrast in horizon­tal in-situ stress, between the layer to be fractured and ad­joining layers has not yet been analyzed in any depth.

Consider an infinitely long fracture of height H in the geometry of Fig. 1, where we now assume that G2 = G 1.

If the horizontal in-situ stress perpendicular to the plane of the fracture is everywhere equal to S 1 , and the fluid pressure, p, in the fracture is uniform, the fracture prop­agation pressure is

1 Kc Pj.h =s 1 + 1.25 .jj{ , ................... (12)

where K c is the critical stress intensity factor or fracture toughness (see Eq. 1).

If, on the other hand, the in-situ stress in the outer layer is equal to S 2 > S 1, the fracture propagation pressure increases to

2 h +(S2 -S 1 )-arccos-. . ............... (13)

11' H The derivation of this expression is simple and is given in Ref. 33. The fracture propagation pressure, P j,h, for a fracture which just reaches the interface at z= ±hl2 is

JUNE 1982

4.0,------------------------,

Fig. 5-Maximum elongation of a hydraulic fracture in a three­layer system as a function of stiffness contrast GiGl' The dashed curve gives the approximate relation

(2L1H}max = 112.J GiGl .

given by Eq. 12 or Eq. 13 with H=h. From Eq. 13 we then have

2 h Pf,H -Pf,h =(S2 -S I) -arccos-

" 11' H

-(l-~) 1.~5 5i This equation gives the extra fluid pressure needed to propagate the fracture into the layer of higher in-situ stress as a function of penetration depth. For the case

Kc 1 MPa'm'/2 (910 psi-in. '/2), G 104 MPa (1.45 x 106 psi), v = 0.25, h 50 m (164 ft), and

S2 -S I 3 MPa (435 psi),

one obtains 10 m (33 ft) of penetration [H=70 m (230 ft)], for an extra fluid pressure Pj,H -Pj,h = 1.46 MPa (212 psi).

The drawback of this analysis, 33 which is the only one we are aware of, is that it is based on a completely static model of fracture propagation. In the example given, it is true that an extra bottornhole pressure of 1.46 MPa (212 psi) will not give a penetration of more than 10 m (33 ft); but the actual penetration must be computed from an analysis of the competition between horizontal and ver­tical fracture propagation, The confining effect of an in­situ stress contrast derives mainly from a reduction in fracture width in the outer layers', which impedes fluid flow and thereby reduces the vertical propagation rate, dH/dt.

Consequently, the effect of an in-situ stress contrast on fracture geometry should be calculated from a coupled elasticitylfluid-flow analysis. An estimate of the effect may be obtained by simple but very approximate calcula­tions, as presented earlier for the case of a stiffness con­trast. A more thorough evaluation requires a three­dimensional (numerical) analysis of the coupled fluid-

347

flow/elastic defonnation problem in and around the fracture. 31,32

Until such calculations have been perfonned, it is not possible to estimate the effect of in-situ stress differences on fracture containment. However, the effect may be quite appreciable. In fact, one of the reasons that shale layers often seem to contain hydraulic fractures may be that, because of their higher ductility or tendency to creep, the horizontal stress in these shale layers often may be higher (closer to the overburden stress) than in adjoining layers. Some direct measurements of in-situ stress seem to confinn this notion. 34

Note that an in-situ stress contrast also may be induced artificially by reducing the pore pressure in the pay zone by partial drainage of the reservoir around the borehole. If the drainage radius is much larger than the height of the pay zone, the resulting defonnation field will have zero lateral strains, while the vertical stress remains equal to the overburden load. From the poroelasticity equations one then finds a linear relationship between change in horizontal in-situ stress in the pay zone, and change in pore pressure

1-2v ,6Shoriz =(l-{3)--,6u

I-v

==')' ,6u, ......................... (14)

where

u pore pressure, (3 ratio of rock grain to rock matrix

compressibility, and v = Poisson's ratio for the fonnation.

The parameter,), usually will have a value between 0.5 and 0.6. Experimental verification of Eq. 14 has been provided by Salz,35 who found a linear relationship be­tween fracture propagation pressure and pore pressure in the Vicksburg field in south Texas; his data scatter around a straight line of slope 0.51.

Note that the Vicksburg field data refer to reservoir depletion over a number of years, in the course of com­mercial production from the field. It recently has been suggested 32 that a massive hydraulic fracture be con­tained by fracturing in stages, allowing time for drainage after each fracturing stage to lower the pore pressure around the fracture. However, since fracturing usually is undertaken in tight (low-penneability) fonnations, the drainage times involved may be excessive; as mentioned earlier, the process works only if the drainage radius is much larger than the height of the pay zone because, by the requirement of stress equilibrium, a local reduction in horizontal (total) stress can be achieved only if part of the horizontal far-field stress is "carried" by shear stresses at the interfaces with cap and base rock. Other­wise, the reduction in pore pressure will be compensated by an increase in horizontal effective stress, the total horizontal stress remaining unchanged.

Conclusions This paper considers the problem of hydraulic fracture containment. It is shown that the concepts of stress inten-

348

sity factor and fracture toughness have only limited ap­plicability in the context of hydraulic fracturing of underground fonnations. As a consequence, local con­tainment effects around the upper and lower crack edge, which decrease the stress intensity factor or increase the fracture toughness, are of only minor importance as regards the ultimate shape of the fracture. Hence, in most cases, the fracture will penetrate into the layers ad­joining the pay zone that is being fractured. However, there is another class (global) of containment effects that tend to limit the penetration depth of the fracture into these layers. These effects are caused by contrasts in stiffness and in-situ stress between pay zone and adjoin­ing layers. For the case of a stiffness contrast, an estimate of the penetration depth has been given. For the case of in-situ stress contrast, the necessary analysis 31,32 has not yet been completed.

Acknowledgment .... This paper is based on work completed during an assign­ment with Shell Development Co., Bellaire Research Center, Houston. Pennission to publish this work is gratefully acknowledged.

Nomenclature D ductility G shear modulus of the fonnation h height of pay zone h' effective fracture height (h 'f), m (ft) H fracture height (hf ), m (ft) K stress intensity factor L fracture (half-) length, m (ft) p fluid pressure, MPa (psi)

Pw fluid pressure in borehole, MPa (psi) ,6p fluid overpressure, MPa (psi)

Q flow rate into (half) the fracture (q), m 3 /s (bbl/min)

r - distance to the crack tip, m (ft) S h horizontal in-situ stress perpendicular to the

fracture u = pore pressure (Pp), kPa (psi)

VV fracture width {3 ratio of rock grain to rock matrix

penneability 1] fluid viscosity (IL), Pa' s (cp) " = Poisson's ratio for fonnation

p s microstructural distance

Subscripts c = critical d drained u = undrained

• Symbols appearing in parentheses here are preferred SPE symbols taken from the "SPE Symbols Standard" published in the 1965, 1968, and 1975 Transactions.

References 1. Perkins, T.K. and Kern, L.R.: "Widths of Hydraulic Fractures,"

J. Pet. Tech. (Sept. 1961) 937. 2. Nordgren, R.P.: "Propagation of a Vertical Hydraulic Fracture,"

Soc. Pet. Eng. J. (Aug. 1972) 306-314.

SOCIETY OF PETROLEUM ENGINEERS JOURNAL

3. Geertsma. J. and de Klerk. F.: "'A Rapid Method of Predicting Width and Extent of Hydraulically Induced Fractures." 1. Pet. Tech. (Dec. 1969) 1571-1581.

4. Geertsma. J. and Haafkens. R.: "'A Comparison of the Theories for Predicting Width and Extent of Vertical Hydraulically Induced Fractures." Trails. AS ME (1979) 101, 8-19.

5. Hilton. P.D. and Sih. G.c.: "'A Laminate Composite with a Crack Normal to the Interfaces." Inri. J. Solids Slru('lures (1971) 7,913-930.

6. Erdogan. F.: "'Fracture Problems in a Non-homogeneous Medium." Conlilll/1II1l Mechallics Aspecrs of' Geodl'/lall1ics and Rock Fracfllrc Mechanics, Thoft-Christensen (cd.). Reidel Publishing Co. Dordrecht. Holland (1974) 45-64.

7. Conminou. M. and Dundurs. J.: "'A Closed Crack Tip Ter­minating at an Interface." 1. App. Mech. (1979) 46, 97-100.

S. Rice. J.R.: "Mathematical Analysis in the Mechanics of Frac­ture." Fracrure, an Advallced Tremise: Vol. /2, Mal/zemalical Fl/ndamellrals, H. Liebowitz (cd.). Academic Press. Washington. DC (1968) 191-311.

9. Rice. 1.R. and Simons. D.A.: '"The Stabilisation of Spreading Shear Faults by Coupled Defonnation-Diffusion Effects in Fluid Infiltrated Porous Materials." 1. Geophvs. Res. (1976) 81, 5322-5334.

10. Sih. G.c.: Handbook of'Slress Inrellsitv Factors. Lehigh U .. Bethlehem. PA (1973).

II. Clifton. R.J .. Simonson. E.R .. Jones. A.H .. and Green. S.1.: '"Determination of the Critical Stress Intensity Factor from Inter­nally Pressurized Thick-Walled Vessels." Ex:p. Me(k (1976) 16, 233-238.

12. Brechtel. C.E .. Abou Sayed. A.S .. and Jones. A.H.: "Fracture Containment Analysis Conducted on the Benson Pay Zone in Col­ombia Well 20538-T," Pmc .. U.S. DOE Second Eastern Gas Shales Symposium. Morgantown. WV (1978) 264-272.

13. Atkinson. B.K.: "Fracture Toughness of Tennessee Sandstone and Carrara Marble using the Double Torsion Testing Method." Inri. 1. Rock Mcch. Min. Sci. and Geoll1ech. Abstr. (1979) 16, 49-53.

14. Abou Sayed, A.S.: "'Fracture Toughness of Triaxially Loaded In­diana Limestone." Pmc., 18th U.S. Symposium on Rock Mechanics. Colorado School of Mines. Golden. 2, 2A3-1 to 2A3-7.

15. Brechtel, C.E.: "Measurements for Mitchell Energy Corporation. Cotton Valley Limestone," reported in Western Gas Sands Project Status Report, U.S. DOE. Morgantown, WV (1978).

16. Schmidt, R.A.: "Fracture Toughness Testing of Limestone." (1976) Expl. Mech. 16, 161-167.

17. Schmidt. R.A. and Huddle. C.W.: "Effect of Confining Pressure on Fracture Toughness of Indiana Limestone," Inll. 1. Rock Me(k Min. Sci. and Geomech. Abslr. (1977) 14,289-293.

IS. Abou Sayed, A.S. and Brechtel. C.E.: "In-Situ Stress Deternlina­tion by Hydro-fracturing: A Fracture Mechanics Approach," 1. Geophys. Res. (1978) 83, 2851-2862.

19. Cleary. M.P.: "Primary Factors Governing Hydraulic Fractures in Heterogeneous Stratified Porous Formations." paper ASME 7S-Pet-47 prcsented at the ASME Energy Technology Conference and Exhibition, Houston. Nov. 1978.

20. Daneshy. A.A.: "Hydraulic Fracture Propagation in Layered For­mations," Soc. Pel. Ellg. J. (Feb. 1978) 33-41.

JUNE 19X2

21. Goodier, 1. N.: "Mathematical Theory of Equilibrium Cracks," FmCIl/re, An Admllces Tre(lti.\·e~Vol. 2, Malhematical FUIl­dalllcl1lals, H. Liebowitz (cd.). Academic Press. Washington. DC (1978) 1-66.

22. Kristianovitch, S.A. and Zheltov. Y.P.: "Fonl1ation of Vertical Fractures by Highly Viscous Fluids," Pmc .. Fourth World Pet. Cong., Rome (1955) 2,579.

23. Biot. M.A.: "General Theory of Three-Dimemional Consolida­tion." 1. Appl. PhI'S. (1941) 12,155-164.

24. Rice. 1.R. and Cleary. M.P.: "Some Basic Stress-Diffusion Solu­tions for Fluid-Saturated Elastic Porous Media with Compressible Constituents." ReI'. Geophys. Space Ph"s. (1976) 14,227-241.

25. Rice. 1.R.: "The Mechanics of Quasi-Static Crack Growth." Proc .. Sth U.S. Natl. Congress of Applied Mechanics, U. of Calif()rnia-Los Angeles (June 1978).

26. Ruina. A.: "Influence of Coupled Defornlation-Diffusion Effects on the Retardation of Hydraulic Fractures." Pmc., 19th U.S. Symposium on Rock Mechanics. U. of Nevada. Reno (1978).

27. Simonson. E.R .. Jones. A.H., and Abou Sayed. A.S.: "Ex­perimental and Theoretical Considerations of Massive Hydraulic Fracturing." Report TR 75-39. Terra Tek, Salt Lake City. UT ( 1975).

28. Schmidt. R .. Northrop. D .. and Warpinski. N.: "Hydraulic Frac­turing Near an Interface: Observations and Calculations Regarding Geometry and Containment," Proc .. 20th U.S. Symposium on Rock Mechanics (1979) 2.

29. Warpinski, N.R., Schmidt. R.A .. and Northrop, D.A.: "In-Situ Stresses. the Predominant Influence on Hydraulic Fracture Con­tainment," 1. Pet. Tech. (March 1982) 653-663.

30. Hanson. M.E .. Anderson. G.D .. and Schaffer, R.J.: "Theoretical and Experimental Research on Hydraulic Fracturing." paper ASME 78-Pet-49. presented at the Energy Technology Con­ference and Exhibition. Houston. Nov. 1978.

31. Clifton. R.1. and Abou Sayed. A.S.: "On the Computation of the Three-Dimensional Geometry of Hydraulic Fractures." paper SPE 7943. presented at the SPE/DOE Symposium on Low­Permeability Gas Reservoirs. Denver, May 20-22. 1979.

32. Cleary. M.P.: "Rate and Structure Sensitivity in Hydraulic Frac­turing of Fluid-Saturated Porous Formations." Proc .. 20th U.S. Symposium on Roek Mechanics. Austin. TX (1979) 137-142.

33. Simonson. E.R .. Abou Sayed. A.S., and Clifton. R.J.: "Contain­ment of Massive Hydraulic Fractures." Soc. Pel. Ellg. J. (Feb. 1978) 27-32.

34. Brechtel. C.E .. About Sayed, A.S., Clifton. R.l.. and Haimson, B.C.: "In-Situ Stress Determination in the Devonian Shales. Ira McCoy 20402. Within the Rome Basin," Report TR 76-36, Terra Tek. Salt Lake City. UT (1976).

35. Salz, L.B.: "Relationship Between Fracture Propagation Pressure and Pore Pressure." paper SPE 6870 presented at the SPE 52nd Annual Technical Conference and Exhibition. Denver, Oct. 9-12, 1977.

SPEJ

Original manuscript received in Society of Petroleum Engmeers office July 31, 1980. Paper accepted for publication Sept. 11, 1981 Revised manuscript received Feb. 26, 1982. Paper (SPE 9261) first presented at the SPE 55th Annual Technical Con­ference and Exhibition held in Dallas Sept. 21-24, 1980.