¨¨WHF*

Widths of Hydraulic Fractures

ABSTRACT

T. K. PERKINS l. R. KERN

MEMBERS AIME

A study of fluid mechanics, rupture of brittle mate­rials and the theory of elastic deformation of rocks shows that, for a given formation, crack width is es­sentially controlled by fluid pressure drop in the frac­ture. Operating conditions which cause high pressure drop along the crack (such as high injection rate and viscous fluids) will result in relatively wide cracks. Conversely, operating conditions which cause low pres­sure drop (low injection rates and thin fluids) will result in relatively narrow cracks.

Charts and equations have been derived which permit the estimation of fracture widths for a variety of flow conditions and for both horizontal and vertical frac­tures.

INTRODUCTION

There has been considerable speculation conoerning the geometry of hydraulically created fractures in the earth's crust. One of the questions of practical im­portance is the width of fractures under dynamic con­ditions, i.e., while the fracture is being created and extended. Such width information could be used, for instance, to help estimate the area of a fracture gen­erated under various conditions. m Also, there has been a recent trend toward the use of large propping par­ticles.",15 Therefore is is desirable to know what factors can be varied in order to assure entry of the large particles into the fracture.

There has been some work on fracture widths re­ported in the literature. In particular, there have been several Russian publications dealing with this sub­ject."'!'''' These papers have dealt principally with the elastic theory and the application of this theory to hydraulic fractures. These studies have not led to an engineering method for estimating fracture widths under dynamic conditions. A recent paper' has reviewed and summarized the Russian concepts.

An earlier paper' from our laboratories also discussed the application of the elastic theory to hydraulic frac­tures. This first approach, based largely on photoelastic studies, has proved to be too simplified to accurately describe the fracturing process. However, these early thoughts have served as a guide during the development of more exact concepts.

We would like to present in this paper our current

Original manuscript received in Society of Petroleum Engineer::; office May 2. 1961. Revised manuscript received July 31. 1961. Paper to be presented at 36th Annual Fall Meeting of SPE, Oct. 8-11, 1861. in Dallas.

lOReferences given at end of paper. SPE 89

SEPTEMBER, 1961

THE ATLANTIC REFINING CO, DALLAS, TEX,

concepts regarding fracture widths and some estimates of hydraulic fracture widths for several conditions. We believe that it is now possible to predict with fair accuracy the factors influencing fracture widths. Furth­ermore, the method of prediction has been reduced to a simple and convenient graphical or numerical calcula­tion.

CRACKS IN A BRITTLE, ELASTIC MATERIAL

Many investigators"":lO have shown that competent rocks behave elastically over some range of stresses. Of course, if the tensile stress imposed upon a rock ex­ceeds some limiting value, then the rock will fail in tension. In similar manner, there are some limiting shear stresses that can be imposed upon rocks. Hubbert and Willis" have discussed the shear conditions which will lead to failure,

Under moderate stress conditions (such as those likely to be encountered when hydraulically fracturing) and when stresses are rapidly appl~ed, relatively, most rocks will fail in a brittle manner. Hence, for this discussion of hydraulic fractures in the earth's crust, we assume the rocks behave as brittle, elastic materials.

Let us develop the discussion in the following way. (The following thoughts are applicable only to brittle materials. )

1. First we consider a brittle, elastic system, An energy balance will show the minimum pressur,e neces­sary to fracture rock, and from this pressure we cal­culate the minimum crack width resulting from ex­tension of a hydraulic fracture.

2. Then we will show that, under ordinary fractur­ing conditions, fracture widths are appreciably greater than the minimum widths of extending fractures. In fact, we will find that crack width is controlled by fluid pressure drop in the fracture.

3. We will discuss pressure drops in fractures and resulting crack widths for various operating conditions and both vertical and horizontal fractures.

4. Finally, we will discuss the significance of these concepts, their relationship to fracturing pressures, etc.

First, consider minimum fracture extension pres­sures. We can shed some light on this question by considering the theory proposed by Griffith"S to ex­plain the rupture of brittle, elastic materials. Griffith recognized that solid materials exhibit a surface energy" (similar to surface tension in a liquid). The funda­mental concept of the Griffith theory is that, when cracks spread without the application of external work (in the interior of an elastic medium which is stressed

937

externally), the decrease in strain energy resulting from the elastic strain in the vicinity of the crack is balanced by the increase in surface energy.

A similar approach when external work is applied by the injection of fluid can be used to estimate hydraulic fracture widths in rocks under static conditions (when the fluid has stopped moving). Consider an infinite elastic medium containing a plane crack bounded by a circle-a penny-shaped crack. If fluid were injected into this pre-existing crack but at a pressure less than that necessary to extend the fracture in length, then the crack would be "inflated". For a perfectly elastic medium, the relationship between crack shape and pressure within the crack has been calculated by Sned­don." Fig. 1 shows a conceptual sketch with some of the geometrical relationships predicted by the elastic theory.

Now let us suppose the pressure in the crack is increased until the crack is just ready to extend in radius. Let volume dV be injected at the fracture ex­tension pressure P, and suppose this results in an in­crease in fracture radius dC. The amount of work done to pump the fluid into the fracture is clearly PdV. The amount of energy stored in the fracture system arises from (1) an increase in potential energy of the elastic medium and (2) an increase in the surface energy caused by the increase in fracture radius. To obtain the minimum fracture extension pressure, the work done is equated to the energy stored in the fracture system. By using this approach, Sack" has derived an equation which will give the minimum pressure necessary to ex­tend a fracture in rock (Eq. 1).

(p _ a) - ~ / 7r a E ) '" - ,,2(1 - v')C (1

where Pm =minimum fracture extension pressure, psi, (J = total earth stress perpendicular to the plane

of the fracture, psi (total earth stress is defined as the sum of stress in rock matrix plus pore pressure, see Ref. 11),

a = specific surface energy of the rock, ft-lb/sq in.,

E = Young's modulus of the rock, psi, v = Poisson's ratio for the rock, and

C = fracture radius, ft. This equation predicts that, for a given rock (i.e.,

fixed values of surface energy, Young's modulus and

W-TOTAl CRACK WIDTH AT RADIUS, r Pm- A UNIFORM PRESSURE ACTING OVER THE

SURFACE OF THE CRACK.

(T. TOTAL EARTH STRESS PERPENDICULAR TO THE PLANE OF THE CRACIC

C • FRACTURE RADIUS

r • RADIUS UNDER CONSIDERATION

II· POISSONS RATIO FOR ROCK

E • YOUMS'S MOOULUS OF ROCK

~211~~-'-''-'-'-~'-1I-'-'-r-r-r.-'-'-'-,,-.

i~~\ I 00111111111111 ttlil

938

-1.0 -0.& 0 O.!i 1.0

(i)

FIG. I-SKETCH OF AN UNRESTRICTED FRACTURE WITH GEOMETRY PREDICTED BY THE ELASTIC THEORY.

Poisson's ratio), the minimum fracture extension pre~­sure (in excess of the earth stress) varies inversely as the square root of the fracture radius.

EXPERIMENTAL VERIFICATION OF THE GRIFFITH-SNEDDON APPROACH

The Griffith-Sneddon approach to minimum fracture extension pressure can be verified experimentally. Sned­don has shown that the volume of a radially symmetr­ical crack with a uniform pressure P acting in the crack is given by Eq. 2.

V = 16 (1 - v') C' (P - a)

3E where V = volume of the crack.

(2)

Combining this with Eq. 1 to eliminate C yields Eq.3.

[ 27r'.a'E' ] 1/'

(p - a) -- 3(1 - v')'V

(3)

Hence, the fracture extension pressure should vary inversely as the fifth root of crack volume.

We have measured fracture extension pressure and crack volume for a Portland cement model. Fluid leak­off into the model could not be eliminated completely. However, it was minimiud by using a fluid of low leak-off properties and a cement of low permeability. The volume of fluid leaking off was estimated and subtracted from the total volume injected to estimate the volume of fluid remaining in the crack. The model was arranged so that the pressure acting within the fracture could be measured.

Fig. 2 shows a sketch of the model and a plot of log P vs log V. A line with slope of "- 1/5" is drawn through the data, thus verifying the relationship shown by Eq. 3.

MIMIMUM FRACTURE EXTENSION PRESSURES

To predict minimum fracture extension pressures for field cases, we must determine typical physical proper­ties of rocks.

RADIALLY SYMMETRICAL FRACTURE INSIPE CEMENT BLOCK

: 1000

z

u h

rC~~~~R J , , ALUMINUM FINS

CEMENTED TO TUBE

SIDE

DETAIL OF FRACTURE INITIATING FINS

rS~OPE OF LiNE' _I~ -4.

'- , , r.-~

" z ... :;

Ii - (P -0-). t2 11"3a 3 E

2] 11 3{J-v'lV

...

... '" => t; ~ 1°8.01

I 0.1

CR~CK VOLUME, ec

I i

! i ,0

FIG. 2-SKETCH OF PORTLAND CEMENT MODEL AND THE

CRACK PRESSURE·VOLUME RELATIONSHIP.

JOURNAL OF PETROLEUM TECHNOLOGY

The values of Poisson's ratios for rocks reported in the literature""ao range from about 0,05 to 0.25. Since the minimum fracture extension pressure is not very sensitive to changes in Poisson's ratio, the use of an average value of 0.15 appears justified.

Young's moduli for rocks vary over a much broader range. Elastic moduli are influenced by the type of rock, porosity and average stress. Table 1 gives ranges of moduli that appear to be in agreement with most literature data. 2

,4,30

There is a paucity of surface energy data for rocks. The U.S. Bureau of Mines" has reported some of the best data. They report a value of 0.0265 ft-Ib/sq in. for the surface energy of quartz crystals and 0.00613 ft-Ib/ sq in. for the surface energy of calcite crystals (CaCO,).

The apparent surface energy during fracturing prob­ably corresponds to the surface energy of the cement­ing material and quartz grains actually fractured. Be­cause of the porous nature of rock, however, the new surface actually created during cleavage is less than 1 sq in./sq in. of fracture surface. Hence, the apparent surface energy during typical fracturing is estimated to be not more than 0.01 ft-Ib/sq in. and perhaps much less.

The minimum pressures needed to extend an average fracture can now be estimated. Let us take a simplified case of an axially symmetrical fracture (a penny-shaped crack). Assume that the following properties are typi­cal: E = 4 X 106 psi, a = 0.01 ft-Ib/sq in. (or less) and v = 0.15.

The minimum fracture extension pressure can be calculated from Eq. 1, and the corresponding maximum crack width at the wellbore can be calculated from Eq. 4. Eq. 4 is derived by Sneddon" for the case of a penny-shaped crack.

W = 8(P - (1') (1 - v')C (4)

m ~E . where W m = maximum crack width at the wellbore.

The calculated values are plotted vs the fracture radius on Fig. 3. These curves show that, for this con­dition of static fluid in a hydraulic fracture, the crack widths are very small.

FRACTURE WIDTHS UNDER DYNAMIC CONDITIONS

From the previous discussion it is clear that under static conditions (and assuming no fluid leak-off) frac­tures would be very narrow. If fluid were injected at rea­sonable pump rates into these narrow cracks, the in­jection pressures would have to be extremely large. However, the resulting high fluid pressures in the frac­ture would force the fracture walls farther apart. As the crack width increased, the pressure necessary to inject the fluid would decrease. In actuality, an equili­brium condition is reached. The resulting fracture width is controlled essentially by the fluid pressure drop in

TABLE l-ESTIMATES OF YOUNG'S MODULI OF FORMATION ROCKS Type of Rock

Porous, Unconsolidated to Lightly Consolidated (Friable) Sands

Medium·Hardness SOrldstone Hard, Dense Sandstone Limestone and Dolomite

SEPTEMBER, 1961

Probable Value of E (psi)

0.5 to 1.5 X 10' 2 to 4 X 10" 5 to 7.5 X 10" 8 to 13 X 10'

the fracture. The fluid pressure at the leading edge of the crack is nearly equal to the opposing earth stress. (The pressure in excess of the earth stress at the leading edge of the crack is usually negligible, as we have just seen in the previous section; therefore, it is taken as zero in the remainder of this paper.) This is shown conceptually on Fig. 4. (A similar sketch could be drawn for a horizontal fracture.) Operating conditions which lead to high pressure drop along the crack (such as high injection rates and viscous fluids) will result in relatively wide cracks. Conversely, operat­ing conditions which lead to low pressure drop (low injection rates and thin fluids) will result in relatively narrow cracks.

Since crack widths are controlled by pressure drop in the crack, we must consider several controlling situations. In this paper, we will discuss the following cases.

1. Vertically oriented, vertically restricted fractures -includes crack widths resulting: (a) from Newtonian fluids flowing along the crack in laminar flow; (b) from Newtonian fluid flowing along the crack in turbulent flow; (c) from non-Newtonian fluids flowing along the crack in laminar flow; and (d) when a large amount of sand in the fracture restricts fluid flow and thereby increases pressure drop along the fracture.

2. Horizontally oriented, axially symmetrical frac­tures resulting from Newtonian fluids in laminar flow­includes: (a) fractures deep within the earth; and (b) very shallow fractures.

There are, of course, other conditions which could be considered. However, we believe that these cases will cover most situations and generally will show the be­havior to be expected. (For instance, widths of un­restricted vertical fractures will be about the same as for horizontal fractures of the same size.)

U) 0.03 Q.

.... E • 4 XI06

PS I U) U a -0.01 Ft. Lb.lin.2 1&1 Z % ....

1/ =0.15 U ~ Z .... ~ 0.025 -~

120 .... 0 ~

0 .... CD

~ j ...J CJ> 100 0.020 i;! (/) .... ~ ~ Q. ....

% Z I-0 U) 0.015 I-Z < .... I- % X I-W Q

.... ~ a:: 0.010 j lI&: I- U U • ..: cr: cr: u

~

::I 0.005 ::I :::I :::I ::I !

z x

'" ::I ::I

b 0 0

~ 0 200

C,RADIUS OF FRACTURE. FT

FIG. 3-MINIMUM FRACTURE EXTENSION PRESSURES AND MAXI-

MUM CRACK WIDTHS FOR A TYPICAL UNRESTRICTED FRACTURE.

939

VERTICAL FRACTURES

Within the earth there often are conditions which will cause vertical fractures to be limited in growth in a vertical direction. Zones with horizontal stress higher than in the pay zone are sometimes found above and below the pay zone and will, for instance, cause the vertically limiting effect. (High horizontal stresses are sometimes found in shale. Also, in limestone reservoirs, nonpermeable sections may have higher horizontal stresses than permeable sections after the reservoir pressure has been drawn down.) The fracture will grow until it reaches the bounding zone and then will be re­stricted in vertical growth. However, it will continue to extend laterally away from the well bore as shown on Fig. 5. The fracture will penetrate up and down into the bounding zones until an equilibrium condition is reached. This equilibrium condition is similar to that of a lever at static equilibrium. The fracturing fluid exerts a pressure against the fracture face tending to open and extend the fracture. The high stresses at top and bottom tend to close the fracture. The fracture, therefore, will extend into the bounding zones until the opposing forces counter-balance one another. If these bounding zones are not thick enough or if pressure drop in the fracture gets high enough, then the fracture may crack through into other zones.

CRACK WIDTHS RESULTING FROM NEWTONIAN FLUIDS IN LAMINAR FLOW

Fortunately, it is rather easy to predict whether a Newtonian fluid will be in laminar or turbulent flow. In a classic work," Reynolds discovered that turbulence would begin when the Reynolds' number (the ratio of viscous to inertial forces in the fluid) exceeded a certain value. In terms of fracturing conditions, the Reynolds' number can be reduced to (see Appendix A) :

SURFACE PRESSURE

THERE IS A HYDROSTATIC PRESSURE DUE TO THE WEIGHT OF FLUID IN TUBING OR CASING

PRESSURE EQUALS EARTH STRESS + PRESSURE DROP IN FRACTURE + PRESSURE DROP THROUGH PERFORATIONS

OF ANY)

THERE IS A PRESSURE DROP DUE TO FLUIO FLOW DOWN TUBING OR CAS ING

FLUID PRESSURE DROP IN FRACTURE

LEADING EDGE OF FRACTURE

PRESSURE ESSENTIALLY EQUALS EARTH STRESS

PRESSURE EQUALS EARTH STRESS + PRESSURE DROP IN THE FRACTURE

FIG. 4--SKETCH SHOWING PRESSURES DURING A FRACTURE JOD.

940

W", IS MA)(IMUM WIDTH OF FRACTURE AT WELL BORE

END VIEW

FIG. 5-SKETCH OF A RESTRICTED VERTICAL FRACTURE (NOT DRAWN TO SCALE).

N = 7.S1 X 10" (m (Sp Gr) /I, (H) (ft)

where Nil" = Reynolds' number, Q = total injection rate, bbljmin,

Sp Gr = specific gravity of fracturing fluid, H = height of fracture, ft and fl, = viscosity of fracturing fluid, cpo

Then, when (Q) (Sp Gr)/(H) (ft) is less than about 0.32, the fluid within the crack will be in laminar flow. The crack width is then given by Eq. 5 (see Appendix A for derivation).

W = 0.38 [(Q) (~) (L)] 1/1 (5)

where W = maximum crack width at the wellbore, in.,

Q = total pump rate, bbljmin, fl, = effective fracturing fluid viscosity, cp, L = length of a vertical fracture measured

from the wellbore, ft, and E = Young's modulus of formation rock, psi.

Using the range of values of Young's moduli from Table 1, Eq. 5 is presented in a convenient graphical manner as Fig. 6.

As an example, suppose conditions for a fracture job in a medium-hardness sandstone were as follows: Q = 30 bbljmin, ft = 4 cp, L = 500 ft, Sp Gr = 0.9 and H = 40 ft.

Q (Sp Gr) = (30) (0.9) = 0 17 Hft (40) (4) ..

(Therefore, fluid is in laminar flow and Fig. 6 can be used.)

QftL = (30)(4)(500) = 6X104•

Therefore, Fig. 6 shows that the maximum crack width at the wellbore would be about 0.13 to 0.15 in.

CRACK WIDTHS RESULTING FROM NEWTONIAN FLUIDS IN TURBULENT FLOW

If (Q)(Sp Gr)/(H)(ft) is greater than 0.32, then the fluid will be in turbulent flow within the fracture. For this case, the width is given by Eq. 6 (see Ap­pendix B for derivation).

W = . [(Q)'(SP Gr) (L) ]'/\ 06 (E) (H)

(6)

where Sp Gr = specific gravity of fracturing fluid, and H = height of a restricted, vertical frac­

ture, ft. Again using values of Young's moduli from Table 1,

Eq. 6 can be shown graphically as Fig. 7. Let us con­sider the same example as before, except suppose that the viscosity of the fracturing fluid were 1 cp, Q = 30 bbljmin, ft = 1 cp, L = 500 ft, Sp Gr = 0.9, and H =

40 ft. Q(Sp Gr)

Hft (30(0.9) = 0.68. (40)(1)

Therefore, the fluid is in turbulent flow and Fig. 7 can be used.

Q'(Sp Gr)L

H (30)'(0.9) (500) ------;-0:--- = 1.01 X 10'.

40 Fig. 7 shows that in this case the maximum crack

width at the wellbore would be about 0.14 to 0.16 in.

CRACK WIDTHS RESULTING FROM NON·NEWTONIAN FLUIDS IN LAMINAR FLOW

If non-Newtonian fluids such as gelled oils or emu1-

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sions are used, then it is necessary to determine the fluids' flow properties before estimating crack width. Fann meter data should be plotted in a manner similar to the log-log plot shown on Fig. 13 in Appendix C. The equation of the straight-line portion is reduced to two constants, k' which is proportional to an intercept at a particular rotor speed, and n' which is proportional

to the slope of the line. (Details for actually deter­mining k' and n' from Fann data are given in Appen­dix C.) These two constants are then used in place of "viscosity".

en

LEGEND:

Unfortunately, we have not been able to plot crack widths for non-Newtonian fluids in as simple a manner as Figs. 6 and 7. It has been necessary to include in the

un-UNCONSOLIDATED TO LIGHTLY CONSOLIDATED SANDSTONE

~-MEDIUM SANDSTONE

~-HARD SANDSTONE

~-LIMESTONE AND DOLOMITE

~ 1.0

USE THiS CHART IF:

(QSPM)(Sp.Gr.) <0.32 (HFTJ ("cp.)

tJ Z

~ tJ <[ a: tJ

u. 0

% l-e ~

:IE :J 2

O. I

It

~ t -- I ~ -

~~ W; ~ -:::::;.:: EXnMPLE ~~ ~'I 1-- .- ,,!!OB~r1 'r 1+- t- ~.J. ~~ ",. -1

~~ ~~~ '...f ~ ?

II

~ 1 I

~- 1 i I,

t I

4 MESH

6 MESH

8 MESH 10 MESH 12 MESH

16 MESH

20 MESH

40 MESH

x .0 I~ I-I~

104 10~

I Iii - 60 MESH

107 ; 10

(0 SPM )(p,cp.)( LFT.)

Q-TOTAL PUMP RATE,SPM

p,-FRACTURING FLUID VISCOSITY, CENTIPOISE

L -LENGTH OF VERTICAL FRACTURE MEASURED FROM THE WELL SORE, FEET

Sp.Gr -SPECIFIC GRAVITY OF FRACTURING FLUID

H = CRACK HEIGHT, FEET

FIG. O--CRACK WIDTHS FOR RESTRICTED VERTICAL FRACTURES RESULTING FROM NEWTONIAN FLUIDS IN LAMINAR FLOW.

LEGEND:

en ~ 1.0 u z

iLZZA-UNCONSOlIDATED TO LIGHTLY CONSOLIDATED SANDSTONE

IZSZSZJ-MEDIUM SANDSTONE

~-HARD SANDSTONE

~ -LIMESTONE AND DOLOMITE

USE THIS CHART IF:

(OBPM )(Sp. Gr.) "--.::..:....:::....:..'--''----'- > O. 3 2 (H FT'> (JLcp.)

~

u. 00. I

1-0--- - - -

~ EXAMPLE PR08LEM .~~~~ ~ ~ --T -

~ ~Wi~ :I: l-e

~

:IE ::l :::E

'7 ~ oK £;;i1~

~--- ~~ I~ 11 IT x .0

<[ I :::E

10

Q =TOTAL PUMP RATE, BPM

fL = FRACTURING FLUID VISCOS I TV, CENTI POI SE

I

1

I +

L = LENGTH OF VERTICAL ':RACTURE MEASURED FROM THE WELL BORE, FEET

Sp.Gr = SPECIFIC GRAVITY OF FRACTURING FLUID

H =CRACK HEIGHT,FEET

10 5

4 MESH

6 MESH

8 MESH 10 MESH 12 MESH

16 MESH

20 MESH

40 MESH

60 MESH

106

FIG. 7-CRACK WIDTHS FOR RESTRICTED VERTICAL FRACTURES RESULTING FRO~l NEWTONIAN FLUIDS l'i TURImLE:'>T FLOW.

SEPTFMHER. 1'161 941

design chart Young's moduli of the formation rocks. Values of Young's moduli can again be estimated from Table 1. After the flow constants k' and n' are deter­mined for the fluid and the Young's modulus of the rock has been estimated, crack widths can be esti­mated from Fig. 8. (See Appendix C for derivation.)

Suppose, for a given fracturing case, k' = 4.69 X 10-2 lb-secn

' /sq ft, r = 0.654, * Q = 20 bbl/min, L = 100 ft, H = 30 ft, and E = 6 X 10' psi (estimated for hard sandstone from Table 1).

k'Q''£ H 1-

n' (4.69 X 10-2)(20)°.654(100)(30)°316

E 6X1~ = 1.8 X 10-5

•

From Fig. 8, the maximum crack width at the well­bore is then estimated to be 0.2 in.

CRACK WIDTHS RESULTING WHEN SAND IN THE FRACTURE RESTRICTS FLUID FLOW

The crack widths estimated from Figs. 6, 7 and 8 apply when pure fluids are being pumped along a frac­ture. These estimated widths are also valid when there is a sparse distribution of propping agent suspended in the fluid (because a dilute suspension of solid material will not appreciably influence pressure drop along the crack). However, if a large amount of sand is injected as a propping agent, then its presence in the fracture will influence pressure drop and thereby influence crack width.

In field operations, sand is usually placed in the fracture such that one of the two following conditions will be obtained: (1) a great deal of sand settles to the bottom of the fracture; or (2) the sand is nearly uni­formly suspended in the fracture and concentrated due to leak-off of the fracturing fluid.

"These particular values of k' and n' are for a very thick gelled fracturing oil.

-" !.

..: ... E ..: ...

..J .. D.

"i UJ

"-

'" £

T g .... .. ;:;: ... -..::=....-

-3 10

-. 10

105

-. 10

107

-8 10

-. 10

. .

~ ~

(THIS EQUATION DEFINES k' AND n~ SEE APPEND IX CI

)---- T Ib F k,(-dV -It - = - sec. ~

Ft.2 d Y

)----t-T'EXAMPLE PROBLEM

- - - -Q= TOTAL PUMP RATE, BPM I L = DISTANCE FROM EXTEND-

I NG EDGE OF CRACK

H • CRACK HEIGHT, FEET

E = YOUNG'S MODULUS FOR THE FORMATION ROCK / I PS I (SEE TABLE II

I !! II II

f;>' 'II o· hj ~ "-

'" o· ~-to~ ~] O-,-",';+: -0-0 , Q)

I I o·

V 1/ 1/

II II II I '/

'/I / II I / 0!D

r ftj ~

.L II 1/1/

7lYw iJ

tlttJ 'I/, ...

~ ... i-

1~ /~I

'" O'

0

I

I

f-

:0;-

~-_ G<D" -10

100.0001 0.001 0.01 0.1 WIDTH,INCHES

i

1.0

Fn;. 8--CRACK WIDTHS FOR RESTRICTED VERTICAL FRACTURE RESTJLTING FROM NON-NEWTONIAN FLUIDS IN LAMINAR FLOW.

The conditions which lead to settling of sand in a fracture were discussed in a pr,evious paper.14 The crack width is not easily calculated for these conditions. We have studied a few cases and found that the crack widths were several times as wide as the predicted crack width without sand present. At the present time, we can only suggest that, for all cases when sand is present, crack width should be estimated in the manner described below where sand concentrates due to fluid leak-off but does not settle.

If the sand is suspended in a very viscous fluid, it will settle so slowly that a settled sand pack will not have time to form during the fracture treatment. How­ever, during the treatment, the fracturing fluid can leak off into the porous formation, leaving a high concentra­tion of sand suspended in the fracture. The viscosity of the concentrated sand slurry will be considerably greater than the viscosity of the pure fracturing fluid. Fig. 9 shows the ratio of the slurry viscosity to the pure fracturing fluid viscosity as a function of the volume fraction of solid material in the slurry."

The crack width is easily estimated in the following way if the sand concentrates because of fluid leak-off.

I, The amount of fluid remaining in the fracture is estimated by the method proposed by Howard and Fast.10 By knowing the amount of fluid in the fracture and the amount of sand in the fracture, the average slurry concentration can be calculated.

2. The average slurry viscosity can then be estimated from Fig. 9.

3. The crack width is then estimated from Figs. 6 and 7 using the average slurry viscosity and density rather than the viscosity and density of the pure frac­turing fluid.

In the actual case, slurry properties vary from point to point in the fracture. Hence, the width calculated as just shown must be interpreted as only an approximate width.

HORIZONTAL FRACTURES

If a fracture is oriented horizontally, crack width may result from two types of rock movement. If the fracture is deep within the earth, crack width results principally from compression of rock in the vicinity of the fracture. However, if the fracture is very shallow, crack width may also result from flexing and lifting of the overburden. This is shown conceptually in Fig. 10 . It is shown in Appendix D that compression of sur-

)- 45 ~

iii 40 0 (,) II) 35 )--

~>

/ /

/ V

/ ./

V

V -:--I-

5

0.1 0.2 0.3 0.4 0.5 0.6 0.7

VOLUME FRACT I ON OF SOLIO MATERIAL IN THE SLURRY

FIG. 9-VISCOSITY OF A SLURRY CONTAINING SUSPENDED SOLID MATERIAL COMPARED TO THE VISCOSITY OF THE BASIC FLUID.

JOURNAL OF PETROLEUM TECHNOLOGY

rounding rock is the principal mechanism leading to crack width if the depth is greater than about three­fourths of the fracture radius. (Hence, this is the mechanism that controls during most fracture treat­ments.) For this condition, the width is given approxi­mately by Eq. 7.

We' ) = 022 [Q(bbl/min)f.L(CP) C(ft)] 1/.' (7) m.. E(psi) .

where C = radius of the fracture, ft. (In deriving this equation we have assumed a homo­

geneous medium. Actually, in the case of a horizontal fracture, the value of Young's modulus should be an effective av'erage for the pay zone and the formations above and below it, since they are also compressed when the fracture is opened. However, for simplicity

SURFACE

OVERBURDEN LIFTED

z

SURFACE

(OVERBURDEN NOT LIFTED APPREC IABLY)

c<~z

ROCK COMPRESSED

FIG. lO-SKETCH SHOWING A SHALLOW AND A DEEP HORIZONTAL FRACTURE.

LEGEND:

we have used the values of Young's modulus for the pay zones as shown in Table 1 in applying this equa­tion.)

Fig. 11 graphically shows the order of magnitude of crack width to expect in horizontal fractures (depth is greater than three-fourths of the fracture radius) if the fluid is in laminar flow. Laminar flow of the fluid at every point in a horizontal fracture is probably en­countered only rarely in field operations. Hence, turbu­lent flow must also be considered before a generally ap­plicable equation can be derived. However, the turbu­lent zone usually will not extend far from the wellbore; therefore, Fig. 11 is approximately correct in all cases.

Appendix D also shows an equation for estimating crack widths of shallow horizontal fractures (depth is less than three-fourths of the fracture radius).

If a large amount of suspended sand is injected into the fracture, then the width of the fracture can be estimated by using the viscosity of the sand slurry rather than the viscosity of the basic fracturing fluid.

CONSEQUENCES OF THESE WIDTH CONCEPTS

Now let us briefly discuss some of the consequences and significance of these width concepts. First, consider the relationship between crack width and fracturing pressure. Pressure drops along the fracture can be esti­mated from Eq. A-lO or D-4 (in Appendixes A and D, respectively) and crack width Eqs. 5, 6, or 7. For those fractures oriented vertically, pressure drops along the cracks will probably range from a few tens of pounds per square inch (for very tall cracks) to perhaps 1,000 psi (for thin zones). The higher pressures in thin zones increase the probability of cracking out of the pay zone into zones above and below.

IZ:ZZ)-UNCONSOLIDATED TO LIGHTLY CONSOLIDATED SANDSTONE

1ZSZS2I- MEDIUM SANDSTONE

aDDD-HARD SANDSTONE

~ - LIMESTONE AND DOLOMITE

(/) I&J :I: o Z

IL. o :I: I­o

I

60 MESH i

I

I .oo~Lo---L-1-1~~~IO~2~-L-L-LLLLUI~03~~~~~WULIO~4~~~-L~~UIO~5--~~~~~1~06S-~~~~~~I07

(Q BPM)(JLcP. )(e FT)

Q. TOTAL PUMP RATE, BPM

JL s FRACTURING FLUID VISCOSITY, CENTIPOISES

C • FRACTURE RADIUS, FEET

FIG. ll-ApPROXIMATE CRACK WIDTHS FOR HORIZONTAL FRACTURES RESULTING FROM NEWTONIAN FLUIDS IN LAMINAR FLOW.

SEPTEMBER. 1961 943

For horizontal fractures or unrestricted vertical frac­tures, pressure drops will be very high initially but quite low for large fractures.

Bottom-hole fracturing pressures will be equal to the sum of the total earth stress perpendicular to the plane of the crack and the pressure drop in the fracture. Hubbert and Willis" and Cleary· have discussed the fac­tors influencing earth stresses.

Fig. 12 shows our estimates of the ranges of earth stresses (similar to estimates of Hubbert and Willis or Cleary) plus pressure drops for vertical and horizontal fractures (as discussed in this paper). This figure also shows some bottom-hole fracturing pressures calcu­lated from actual field treatments. It is apparent that bottom-hole pressures for most of these fracturing treatments fell in the range predicted for vertical frac­tures. It is interesting to note that predicted pressures for vertical and horizontal fractures overlap at depths less than about 3,000 ft. Hence, at these shallow depths, fracture orientation cannot be determined by cursory inspection of the bottom-hole fracturing pres­sure gradient.

Another interesting consequence of these concepts is the behavior to be expected after pumping has ceased. Since the fracture extension pressure is nearly equal to the earth stress, the fracture will continue to extend after pumping has ceased. The crack will become nar­rower and longer until the minimum width (previously discussed) is reached or until formation walls grip the propping material. Of course, fluid will continue to leak-off over the whole fracture area and particularly near the extending edge of the fracture as the walls close. This limits the additional length obtained after pumping stops.

Finally, let us review the factors which influence crack width (see Eqs. 5, 6, and 7).

1. The thickness of pay zone should have no effect on pressure or width of horizontal fractures. For re­stricted vertical fractures, pressure drop along the fracture will be large for thin zones but small for thick zones. On the other hand, crack width at the wellbore is nearly independent of height of a vertical fracture (except in turbulent flow).

2. Depth of pay zone will generally have little effect on crack width (except that rock properties may vary

(f)

'"

IOr------r------r------r------.-~~7r~~_.

~ e~----~------~----~--~ rn

"'­n::_ n.(f)

n.

'" ~~ ::l 0-'" uc <[z n::<[ ... rn

::l ",0 ..J:I: 00-::t:-

2 o 0-0-o III

94·t

DEPTH ,(THOUSANDS OF FEET)

FIG. 12-THEORETICALLY PREDICTED BOTTOM·HoLE

FRACTURING PRESSURES AND FIELD DATA.

with depth). For very shallow horizontal fractures, the width may be somewhat greater than normal because of lifting of the overburden. However, this should be of no significance in most normal fracturing operations.

3. Crack width is not particularly sensitive to rock properties. Young's moduli of rocks have a range of about ten- or twenty-fold. However, crack width is in­versely proportional to the fourth root of Young's modulus; therdore, only about a twofold variation in crack width should be expected from this range of moduli.

4. Since the viscosity of a fracturing fluid (or effec­tive viscosity if slurries are considered) can be varied over a very wide range, this factor will have an ap­preciable effect on crack width.

5. Pump rate will also influence width, but usually the range of pump rates is limited by the horsepower avail­able.

6. Fracture width at the well bore will also be in­fluenced by length or radius of a fracture. As the vol­ume of fluid in the fracture increases, the crack width will increase.

7. Crack width is strongly influenced by a large amount of solid propping material in the fracture. The solid material increases resistance to fluid flow and re­sults in a wider crack.

NOMENCLATURE

C = fracture radius Dc = equivalent diameter E = Young's modulus of rock f = friction factor

H = height of a vertical fracture k' = a measure of the flow properties of a non­

Newtonian fluid. It is determined as ex-

. [(lb)(SeC)"'] plained in Appendix C, -sq-ft----

L = length of a vertical fracture (measured from the well bore )

n' = a measur,e of the flow properties of a non­Newtonian fluid. It is determined as ex­plained in Appendix C.

Nil, = Reynolds' number P = pressure

P,,,." = average fluid pressure in the fracture P'" = minimum fracture ,extension pressure P". = fluid pressure at the wellbore Q = total injection rate into a well r = radius

r w = radius of the well bore R" = hydraulic radius

Sp Gr = specific gravity of fracturing fluid u = velocity v = velocity V = volume W = crack width

W" = average fracture width W", = maximum crack width at wellbore

x = distance y = distance Z = depth of fracture (measured from surface

of ground) (l' = specific surface energy of rock

10 darcies . (3 = (a constant), see Eq. D-I

12 sq cm y = a proportionality constant relating equiva-

JOURNAL OF PETROLEUM TECHNOLOGY

lent diameter and hydraulic radius (see Eq. A-3)

8, = upward deflection of an elastic plate clamped at the edge (see Eq. D-8)

8, = downward deflection of an elastic medium (see Eq. D-9)

f1. = viscosity of a fracturing fluid v = Poisson's ratio for rock p = fluid density o' = total earth stress perpendicular to the plane

of the fracture T = shear stress

REFERENCES

I. Barenblatt, G. 1.: "On Equilibrium Cracks Formed in Brittle Fracture", Soviet physics-Doklady., (1960) 4.

2. Birch, Francis: Editor, Handbook 0/ Physical Constants, GSA Special Paper No. 36 (Jan. 31, 1942).

3. Khristianovitch, S. A. Zheltov, Y. P., Barenblatt, G. I. and Maximovich, G. K.: "Theoretical Principles of Hydraulic Fracturing of Oil Strata", Proc., Fifth World Pet. Cong., N. Y. (1959) Section II.

4. Cleary, J. M.: "Hydraulic Fracture Theory, Parts I, II, III", Circulars 251, 252, 281, Illinois Geological Survey, Urbana, II!. (1959).

5. Davies, S. J. and White, C. M.: Proc., Roy. Soc. of Lon­don (1928) 119A,92.

6. Gilman, J. J.: "Direct Measurement of the Surface Ener­gies of Crystals", Jour. Appl. Phys. (1960) 31., 2208.

7. Griffith, A. A.: Phil. Tra't.I. Roy. Soc of London, (1920) A, 221, 163.

B. Griffith, A. A.: Proc .. Int. Congo App!. Mech. (1924) Delft, 55. '

9. Harrison, Eugene, Kieschnick, W. F., Jr. and McGuire, W. J.: "The Mechanics of Formation Fracture Induction and Extension", Trans., AIME (1954) 201, 252.

10. Howard, G. C. and Fast, C. R.: "Optimum Fluid Char­acteristics for Fracture Extension", Drill. and Prod. Pmc., API (1957) 26l.

11. Hubbert, M. K. and Willis, D. G.: "Mechanics of Hy­draulic Fracturing", Trans., AIME- (1957) 210, 153.

12. HUtIt, J. 1.: "Flui<l Flow in Simulated Fractures", AIChE Jour. (1956) 2,259.

13. Huitt, J. L. and McGlothlin, B. B.: "The Propping of Fractures in Formations in which Propping Sand Crushes", API Paper No. 875-13-E (May, 1959).

14. Kern, 1. R., Perkins, T. K. and Wyant, R. E.: "The Me­chanics of Sand Movement in Fracturing", Trans., AIME (1959) 216, 403.

15. Kern, 1. R., Perkins, T. K. and Wyant, R. E.: "Propping Fractures with Aluminum Particles", Jour Pet. Tech. (June, 1961) 583.

16. Lamb, Horace: Hydrodynamics, Sixth Ed., Cambridge U. Press (1932).

17. Moody, L. F.: Trans., ASME (1944) 66, 671. 18. Perry, J. H.: Editor, Ch!emical Engineers' Handbook, Third

Ed., McGraw Hill Book Co., Inc., N. Y. (1950) 378. 19. Reynolds, 0.: "An Experimental Investigation of the Cir·

cumstances Which Determine Whether the Motion of Water Will Be Direct or Sinuous, and of the Law of Re­sistance in Parallel Channels", Phil. Trans. (1883) Roy. Soc. of London.

20. Rothfus, R. R., Archer, D. H., Klimas, I. C. and Sikchi, K. G.: "Simplified Flow Calculations for Tubes and Paral­lel Plates", AIChE Jour. (1957) 3, 208.

21. Sack, R. A.: "Extension of Griffith's Theory of Ruptur.e to Three Dimensions", Proc., Phys. Soc. of London (1946) 58, 729.

22. Sneddon, 1. N.: "The Distribution of Stress in the Neigh­bourhood of a Crack in an Elastic Solid", Proc., Roy. Soc. (1946) A, 187, 229.

23. Sneddon, 1. N. and Elliott, H. A.: "The Opening of a Griffith Crack Under Internal Pressure", Quarterly of Appl. Math. (1946) 4,262.

24. Stein our, H. H.: "Rate of Sedimentation - Nonfloccu-

SEPTEMBER, 191>1

lated Suspensions cf Uniform Spheres", Ind. Eng. Cheni. (1944) 36, 618.

25. Timoshenko, S.: Theory of Plates and Shells, McGraw­Hill Book Co., Inc., N. Y. (1940) 60.

;~6. USB:\f Bulletin 402, U. S. Dept. Interior. 27. Walker, J. E., v/han, G. A. and RothfllS, R. R.: '-Fluid

Friction in Noncircular Duets", AIChE Jour. (1957) 3, 484.

28 Way, S.: "Bending of Circular Plates \I'ith Large De­Oection,,", Trans., AIME (1934) 56, 627.

29. '\'/eil, N. A. and Newmark, N. ;-'1.: "Large DeOections of Elliptical Plates", Jour. Appl. Mech. (1956) 23, 21.

,~(). Wllcrker, R. C~,: "Annotated Table,; of Strength and Elastie Properties d Rocks", Paper 66.'3-G published as separate publication by SPE 09S61.

:It. Zheltov, Y. P. and Khristianovich, S. A.: "The Hydraulic Fracturing of an Oil-Producing Formation", izvest, Aklld. Nallk USSR, Otde!' Tckh. Nauk (1955) No.5, 3.

32. Zhelto\" Y. P.: "An Approximatc :\Iethod of Calculating the Size of Fissures Produced hy the Hydraulic Fracturing 01 :l Formation", Izvest. Akad. ,iVauk USSR. Otde!' Te!dl. Nauk (957) No . .'3, 180.

APPENDIX A

DERIVATION OF AN EQUATION GIVING VERTICAL CRACK WIDTHS RESULTING FROM

NEWTONIAN FLUIDS IN LAMINAR FLOW

Widths of restricted, vertical cracks resulting from injection of Newtonian fluids can be estimated by mak­ing the following assumptions.

I. Assume that the fracture is vertical and of fixed height H (the height is independent of the distance from the well bore ) .

2. Assume that the cross-sectional shape of the crack at any point is essentially elliptical and that the maxi­mum width at that point is proportional to the difference betw,een pressure and stress at that point. The equation of the ellipse is taken as Sneddon's equation for a two­dimensional system (Eq. A-lO).

3. Assume that the pressure drop can be estimated by means of Fanning's equation in which the hydraulic radius for an ellipse has been substituted.

4. Assume that the fluid pressure at the extending edge of the crack is essentially equal to the total earth stress perpendicular to the plane of the fracture.

5. Assume there is no leak-off of the injected fluid. (By assuming the limiting condition of a total fluid leak-off rate large enough that the fracture is no longer extending, one can readily show that leak-off has little effect on crack width.)

The Fanning equation may be written as Eq. A-I. dP 2 f v' p

dx Do where, for laminar flow,

f=~~·· D,yp

(A-I)

(A-2)

For an ellipse of essentially zero eccentricity (ec­centricity is the ratio of minor to major diameters), Perry" has shown •

D = yR[[ = y(~) (A-3) , 2.546

where Do = equivalent diameter, y = a proportionality constant,

RIf = hydraulic radius, and W = minor diameter of the ellipse (equals

maximum crack width).

'H5

Lamb'• has shown for an ellipse of nearly zero ec­centricity and for laminar flow

dP 32Qp. dx = -rrH W·: . (A-4)

Substituting Eqs. A-2, A-3 and A-S into Eq. A-I and equating to Eq. A-4 shows

y = 3.6 . (A-5) (This can be compared with y for a circular pipe of 4 or y for parallel plates of 3.26.)

We assume that y = 3.6 is the value that should be used for all cases considered here, both laminar and turbulent. The Reynolds' number of the fluid flowing down the crack can now be written.

( RH

) (flOW rate) N R, = D 6 V P = Y area P

p. P.

( area ) ( flow rate)

y wetted perimeter area P

fl·

3.6( flo;;ate) P

p.

= 7.S1 X 10" (Q) (Sp Cr) (H) (p.)

(A-6)

Fluid will be in laminar flow in the crack if Reyn­olds' number is less than about 2,500. (See, for in­stance, the data of Huitt" or Rothfus, et al20.) Substi­tution of N R6 < 2,500 into Eq. A-6 shows that laminar flow will obtain if

(Q) (Sp Cr) 032 (H) (p.) <. .

In the laminar region, the friction factor is given by Eq. A-2. We can compare this friction-factor relation­ship for parallel plates with experimental data which have been reported in the literature. For smooth paral­lel plates, the data of Huitt,l2 Davies and White: Roth­fus, et aI," and Walker, et aI," are in agreement with each other. Furthermore, in the laminar region the ex­perimental data are in exact agreement with this theo­retical approach.

Substituting Eqs. A-2, A-3 and A-5 into Eq. A-I yields Eq. A-7.

dP 32 (2.546), v p. dx (3.6)' W'

But we note that 2Q

v= -rrWH'

Therefore, dP 64 (2.546)' Qp. dx 7r (3.6)' W'H

(A-7)

(A-S)

(A-9)

For a fracture restricted at top and bottom, Sned­don" has shown

W 2 (1 - v')(P. -I(]') H (A-lO) z= E

If we assume that Q. is constant (this neglects leak­off and accumulation in the crack), then substituting Eq. A-I0 into Eq. A-9 gives Eq. A-It.

_ 3 = [(S)(2.546)'][ p.QE3

]dX (P 0') dP -rr (3.6)' H' (1 - v')"

(A-ll )

946

By integrating and noting that (P - 0') = 0 at the leading edge of the crack, we can express the pressure as a function of the distance from the leading edge of the crack. By then substituting into Eq. A-lO, the width is given as a function of the distance from the end of the crack. Assuming v = 0.15, we can therefore express the width in convenient units as

(. ) = 0 3S [Q(bbI/min) p.(cp) L(ft)]':' (A-12) W m.. E(psi)

APPENDIX B

DERIVATION OF AN EQUATION GIVING VERTICAL CRACK WIDTHS RESULTING FROM

NEWTONIAN FLUIDS IN TURBULENT FLOW

The basic assumptions for this case are the same as those listed in Appendix A. If Reynolds' number is greater than 2,500, we assume that

f = 0.0125 . (B-1) The friction factor is influenced by the r:.mghness of

the walls; the f = 0.0125 value corresponds to a rela­tive roughness" of about 0.02. The effect of relative roughness is clearly shown by the experiments of Huitt."

Substituting Eqs. A-3, A-5, A-S, A-lO and B-1 into Eq. A-I yields Eq. B-2.

(P_ )3dP=[(0.0125)(2.546)]( pQ2E" ]dX 0' . -rr' (3.6) (1 - V

2)3 Ir

(B-2) By integrating, noting that (P - 0') = 0 at the lead­

ing edge of the crack and substituting into Eq. A-lO, we obtain Eq. B-3 for maximum crack width at the wellbore.

_ [(Q)' (Sp Cr) (L) ]'/' W - 0.6 (E) (H) . (B-3 )

APPENDIX C

DERIVATION OF AN EQUATION GIVING VERTICAL CRACK WIDTHS RESULTING FROM

NEWTONIAN FLUIDS IN TURBULENT FLOW

The relationship between shear stress and shear rate for many non-Newtonian fluids (as well as Newtonian fluids) can be r:epresented by Eq. C-l.

( d)'" T = k' - d~ . .

where T = shear stress, u = velocity = fey),

k',n' = constants, and

(~;) = shear rate.

(C-l)

The shear behavior of most fracturing fluids, in fact, can be represented by this question. The constants k' and n' can be evaluated with a Fann meter. Table 2 shows the relationship between shear rate in seconds-' and rotor rpm for a Fann meter.

At each shear rate, the meter will indicate a scale reading which can be converted to shear stress in pounds per square feet by means of Eq. C-2.

(scale reading) (spring constant) (C-2) Tlb/sq ft = 100 .

Shear stress should then be plotted vs shear rate on

JOURNAL OF PETROLEUM TECHNOLOGY

log-log paper. Fig. 13 shows typical data for a thick fracturing oil which has been gelled varying amounts.

A straight line is drawn through each set of data points. (The points at low shear rate frequently deviate from the straight line and should not be used.) The constant n' is then determined from any two points on the straight lines, as indicated by Eq. C-3.

[

(log) (T,/T,) 1 n' = (log) (shear rate) 2 •

(shear rate) 1

(C-3 )

Having 11', the constant k' is determined at any point on the straight line by means of Eq. C-4.

k' = T (shear rate),,'

(C-4)

Now l,et us derive an equation giving crack widths in terms of the constants k' and n'. Fig. 14 is a sche­matic representation of a fluid flowing in laminar flow between parallel plates. For steady, non accelerated flow, the force per unit height acting to the right on the fluid element of thickness y and length L may be equated to the force per unit height acting towards the left as shown by Eq. C-5.

yb.P=LT (C-5) Eq. C-1 may be substituted into Eq. C-5, and the

equation can then be integrated by observing that the velocity is zero at the wall. Eq. C-6 gives the resulting velocity profile.

II = ( n' 11~ 1 )( ~~ ) ~[( ~)n'n:~ _ y n', :1] . (C-6)

The total flow rate into the two sides of the fracture is given by Eq. C-7.

wl2

Q=4 f Rudy

o

2n' +1

=4 ( n' )( b.P )n' .( W)---';;-

211' + 1 k'L H 2 . (C-7)

Re-arrangement of Eq. C-7 yields Equation C-8 ex­plicit in the pressure gradient.

( - ~~) = 2k'Qn' el1'l1~ 1 f ~n' W':'+1' (C-8)

Lamb13 has shown that the pressure gradient when flowing Newtonian fluids in an elliptical conduit of

. 11 .. . (16) . essentIa y zero eccentncity IS 37T tImes as great as

the pressure gradient when flowing the same fluid be­tween parallel plates under the same condition of height, maximum width and flow rate. If it is assumed that this same relationship holds approximately for non­Newtonian fluids, then Equation C-9 applies to es­sentially elliptical fractures.

( _ dP) ~~2k'( 211' + 1 )"'_1_ ~

dx ellipse 37T n' H nl w:!nl+l

Speed of Rotor (rpm)

-~--

300 200 100

6 3

TABLE 2

Rate of Shear ~~~n_d~~.l_

959 479 320 160

9.6 4.8

(C-9)

By substituting in the Sneddon equation for the width of a two-dimensional elliptical crack (Eq. A-lO) and by assuming that the flow rate is independent of x, Eq. C-9 can be integrated. Substitution of the resulting (P - a) into Eq. A-IO yields Eq. C-lO.

= [2(1 - v2)H] { (2n' + 2) cpE''''+1Qn'L}2n'+2 W E [2(1 - v') Hr'+1

(C-lO)

where ,-I-- = ~ k' (~~)'" _1_ . 'I' 37T n' Hn'

Equation C-10 can be written with a more conven­ient set of units as shown by Eq. C-l1.

10

l1

/' ./

~ ~v V ,/ (0'

(O~+:[(/ ,/ Vx ~<i-<":~ ,,'/ / .'''~~''~~ / ~ "~ V (0 ~ / '/, 0 ,,"v'

~:t::?~/v ~,,/ / /0 ,"/ Y

L A /

/ oC+<"/ / +0/

/ x/ /

'" -" ..ci

E ~ry

1.0 (Jl

W 0: I-Ul

0: <t W :c Ul

/ 200 500 1000

SHEAR RATE, SEC.-I

FIG, 13---FA.'1N ,METER DATA FOR GELLED FRACTURING Of!"

/( ~ f{ ~ //r------~--~--~--~~~

:/:/ '" m=1" - - - - - - - ,-","---{--"<"'"

" - - ,,'~ I )---------~~

I I I I I I

% PI t1 I I 'JJ{P2 // +...f I 0( L ~ 1 1// I I 1///

!jl I I I

~ I : '- I I / 17 ~---------~

FIG. 14-LAMINAR FLOW OF FLUllJ BE'I'WElcN PARALLF.L PLATES.

9 t,7

w = 12 [( 32~ ) (/1' + 1) ( 2n'n~ f (O.~::S)( S~~~f r'+'[Qn'k'~ HI~r~~'

(C-ll )

where W = maximum crack width at the well bore, in., k' = a measure of the flow properties of a non­

Newtonian fluid, (lb) (sec)"' /sq ft, n' = a measure of the flow properties of a non­

Newtonian fluid, Q = total injection rate, bbljmin, L = length of a vertical fracture measur,ed

from the well bore, H = height of a vertical fracture, ft, and E = Young's modulus of the formation rock,

psi. Eq. C-II is plotted as Fig. S.

APPENDIX D

DERIVATION OF AN EQUATION GIVING APPROXIMATE CRACK WIDTHS FOR HORIZONTAL FRACTURES

Let us derive an approximate relationship which will give the order of magnitude for horizontal crack widths if the fluid is in laminar flow. Assume that there is some average crack width in the fracture and that this average width can be used to get the approximate pres­sure distribution. Then for a radial system and fluid in laminar flow, the pressure distribution is given by Eq. D-l.

r QiJJn~

rw P = P w - 271: {3 W

a3

(D-l)

wherc P = pressure at radius r, P" = pressure at the wellbore, 1'" = radius of the wellbore, Q = total injection rate, /_t = viscosity of the fluid, {3 = 10'/12 darcies /sq cm (a constant), and

Wa = average craek width. The average pressure exerted over the fracture face

is given by Eq. D-2. G

71: r,/ P w + J 271: r P dr

P avg = 7I:C"

(D-2)

Substituting Eq. D-l into Eq. D-2 and integrating yields Eq. D-3.

Payg = P n - Qp. -[~ln~ + ~(~)' -~] 7I:{3 W,i' 2 r" 4 C 4

(D-3 )

Sneddon" shows that, for a three-dimensional radi­ally symmetrical crack, the width is given approxi­mately by Eq. D-4.

W = S(1 - v') C (PaY. - IT) Tllax "E (D-4)

Based on volume, the average width is related to maximum width by Eq. D-S.

i(2C)' W mnx = 7I:C' W, v",

948

')

WaY.l!"=iWmn:\.. (D-S)

By noting that the pressure at the leading edge of the crack is equal to the earth stress, substitution of Eqs. D-l, D-3 and D-S into Eq. D-4 yields Eq. D-6.

_j27 (1 - v') CQ}J-[l - (~)' 1(.1' Wm " - ) ~_. , _____ ~ (W-6)

, 470' Ef3 I

The termll - (~)' ] may be assumed equ:ll to unity

for any practical size of fracture. Assuming v = O.IS, Eq. D-7 gives the approximate crack width at the well­bore with a more c:mvenient set of units.

W(in.) = 0.22[Q(bbljmin) }J-(cp) C(ft)]'!' E(psi)

(D-7)

This derivation assumes that the crack width results from compression of rock in the vicinity of the frac­ture. If the radius of the fracture is large compared to the depth, then it will be possible to lift the overburden as well as compress the rock near the fracture. Let us now consider crack widths if the overburden is lifted.

Again it is assumed that there is an average pressure acting in the fracture given by Eq. D-3. The deflection of an elastic plate which is clamped at the edges has been given by Timoshenko," Way," and Weil and New­mark." If the center deflection of the plate is less than 40 per cent of the plate thickness, then the deflection is given by Eq. D-S.

8 = 3(P,,," - IT) (1 -v') c'l 1 - (-z.-), r 1 16 EZ'

.(D-S) where 8, = upward deflection of the plate at radius r,

and Z = thickness of the plate (or depth of the frac­

ture) .

The rock below the fracture is compressed slightly; the down-ward deflection is given approximately by Sneddon's equation, Eq. D-9.

8, = 4(Payg - :) ~l - v') C[ 1 _ (~)' f' (D-9)

The resulting maximum crack width at the well bore is given by Eq. D-IO.

W = (PaY. - IT)(l - v') C[ 4 3 (C)3J max E -;:- + 16 Z

Based on volume, the average width can be to maximum width as shown by Eq. D-ll.

c Wayg J 0271:r W dr

WmHX 7T'C2 W IIIH :>.::

(D-lO)

related

(D-ll)

By noting that the pressure at the leading edge of the

JOURNAL OF PETROLEUM TECHNOLOGY

crack is essentially equal to the earth stress, substitution of Eqs. D-J, D-3 and D-Il into Eq. D-IO yields Eq. D-J2.

W mux =:::

( 0 ((rw '][4 3 (C)']')'/' )(1 - vo) C QfL 1 - c) --; + 16 Z ~

~ 3h E 13[3: + 312 (~)T ~

(D-12)

This gives a crack width equal to that calculated for deep horizontal fractures (Eq. D-6) when C/Z;:::;: 4/3.

SEPTEMBER, 1961

Hence, Eq. D-12 is applicable if C ~ 4/3 Z. Neglect­

ing the term [1 - (I~ )'] , assuming v = O. 15 and sim­

plifying units results in Eq. D-J3. Wm",(in.) =

_ )Q(bbl/min) fL(CP) C(ft)[~+-&(~rrr' 0.076) ) [ 4 ()"J ' ( ~ E(psi) 37T + 3

12 ~" )

(D-J3) C 4·

where - > --. Z -- 3 ***

949