4.Paper for Chapter 5_SPE 56473 Skin Bypass Fracturing

8/18/2019 4.Paper for Chapter 5_SPE 56473 Skin Bypass Fracturing

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2 PHIL RAE, A. N. MARTIN AND B. SINANAN SPE 56473

field results1,2

clearly demonstrate that this is certainly not the

case. Indeed, worldwide, an enormous proportion of wells,both old and new, could benefit from the process of hydraulic

fracturing. Before this is likely to happen, however, there will

need to be a rationalization of fracturing methodology to

eliminate many of the myths associated with the technique. To

put it bluntly, we must differentiate between those fracturing

operations where size is truly important and those where,within certain limits, size really doesn’t matter. To this end,

we can start by categorizing fracturing into three main types :

a) Massive Hydraulic Fracturing – this type of operation is

predominantly carried out in tight gas reservoirs wherethe matrix permeability of the formation is low, in many

cases in the range of microdarcies. In order for such wells

to produce at economic rates, the inflow surface must be

increased to thousands, or even tens of thousands, of

square feet. The only way to achieve this is by creating a

fracture and pumping a substantial volume of proppantinto it. The conductivity of such a fracture need not be

particularly high since, even at modest values, thefracture’s ability to transport gas to the wellbore is many

times greater than the exposed formation’s ability to

deliver gas into the fracture. Thus, the fracture need not be

wide but must be long (several hundreds of feet) to

maximise the inflow surface area. To this end, the

treatments involve pumping low concentrations of

proppant for prolonged periods of time, usually several

hours. The total proppant mass typically ranges from

several hundred thousand pounds up to several millions of

pounds ie. 100 – 500+ tonnes.

b) High Permeability Fracturing / FracPacking / TSO

Fracturing – as one of these names indicates, the second

type of operation is conducted, predominantly, in high

permeability oil reservoirs where permeability can range

from a hundred millidarcies up to several darcies. In some

cases, these formations are weak or poorly consolidated.

In such cases, the primary issue is again to provide a

highly conductive path for reservoir fluids from the

formation to the wellbore. However, because of the

already-high formation permeability, there is a limit to the

amount of fluid that even the most conductive fracture can

realistically handle. Thus, the emphasis here is to

maximize the fracture conductivity3,4 while balancing this

with the inflow surface created in the operation. From a

practical standpoint, this demands the creation of a wide

fracture (for maximum conductivity) but one that is

relatively short (several tens of feet). Operationally, these

treatments involve the pumping of high sand

concentrations for a short period, typically 30 minutes to

one hour. Despite the short duration, the total proppant

mass is not insignificant due to the high average

concentrations throughout the treatment. It can range from

50,000 – 250,000 lbm (ie. 20 – 100 tonnes). Ideally, the

intention is to undertake a controlled “tip screen-out”

(TSO) whereby the fracture is systematically packed with

an escalating quantity of proppant. The increase in net

pressure caused by the arrested growth of the fracture and

reducing leak-off, causes the fracture to “balloon”

(depending on rock compliance, which is high in soft,

highly permeable formations). This in turn allows

injection of additional proppant, effectively increasing

width and, thereby, conductivity.c) Skin-Bypass Fracturing (SBF) – the objective in this type

of operation is to create a small fracture extending

through, and beyond, any area of damage in the near-

wellbore. Such damage5 (“skin”) drastically reduces the

ability of the reservoir to deliver fluid to the wellboreunder Darcy radial flow conditions, resulting in impaired

productivity and abnormally high drawdown. Even a very

short, moderate conductivity fracture can have a

tremendous impact on well performance in a damaged

well. Indeed, tangible benefits can still be realised in

situations where skin is negligible, due to the increase ineffective wellbore radius6 resulting from such a fracture.

Such treatments are beneficial even if the fracture doesnot extend across the entire height of the producing

interval. This type of fracturing operation is an alternative

to matrix acidizing. The short depth of penetration and

limited vertical extension make SBF’s extremely cost-effective. The operation itself involves the pumping of a

small volume of sand at relatively low rates. Thus,

equipment requirements are minimal and the job is simple

to execute. Typical quantities of proppant are in the range

of 1000 – 25,000 lbm (0.5 – 10 tonnes) and horsepowerrequirements vary from 400 – 1000 HHP. It must be

emphasized that high quality ceramic proppants should be

used in such operations, whenever possible. Several

authors have demonstrated the tangible benefits of thehigher permeability of such proppants. Given the very

small incremental cost incurred by using ceramics insteadof sand, their use is justified.

It may seem surprising that SBF’s require so little in the

way of hydraulic horsepower. This is a direct consequence of

the low rates (< 10 bpm or < 1.5m3 /min) and lower surface

treating pressures that typify these jobs, although excessive

spurt loss can increase the HHP requirements. The lower

pressures are a direct result of the low injection rates and the

associated reduced levels of tubing friction pressure. The low

rates, in turn, are a consequence of the very limited extent of

these fracture treatments.

Hydraulic fractures are held open by the fluid pressureinside the fracture and this pressure is dependent on the leak-

off rate, the rate of fracture growth and the fluid injection rate.

A large fracture has a large leak-off surface and loss of fluid

from within the fracture can be substantial. Also, fracture

volume increase due to growth at the edges is proportionately

greater than in a small fracture. The combined effects of these

phenomena require that fluid be injected at high rates in larger

scale treatments so that net pressure within the fracture is






simulators. Unfortunately, due to a variety of reasons, neither

simulator was able to model the effects of a fracture whichonly penetrated part of the zone height.

Initial simulation studies concentrated on the unfractured

productivity index the reservoir, to give J for a number of

unfractured zone heights, H – h f at differing values of skin

factor, S . Also, simulations were run at the full zone height H ,

with S varying from 0 to 25, to give the initial, unfracturedreservoir productivity, J 0.

Once these values had been obtained, the study went on to

consider the fractured reservoir. This involved placing a

fracture in the reservoir, with an h f equal to or less than H .

Once the production from the fracture was obtained, it wascombined with the production from the unfractured part of the

reservoir section to give an overall J for the entire interval.

This was then divided by the original J 0 for that particular

value of S to give the folds of increase in productivity, J / J o.

The quantities that were varied during the simulations were

skin factor, reservoir permeability, fracture height, aspect ratioand fracture conductivity. Aspect ratio, AR, is defined as

follows;

AR = x f

h f .....................................................(1)

For most of the simulations this was fixed at 0.5, which gives

a radial fracture. This is a valid assumption, given that a

relatively small fracture in a larger, uniform reservoir section

is being modeled. However, for completeness, the aspect ratio

was varied to illustrate it’s effects. Nevertheless, it is likely

that with most fractures of this type, AR will be close to 0.5.

Fracture conductivity, F c is defined as follows;

F c = k f . w f .....................................................(2)For the purposes of this study, the proppant was considered

to be 20/40 Colorado Sand , with a k p of 273 darcies. The

actual volume of the proppant was using the following

equation9;

V prop =124.6 F CD x f

2 k f h f γ (1-φ p)

k p ...................(3)

Most of the simulation work was carried out on a

hypothetical oil reservoir. Once clear trends had been

established, this work was extended to cover a base case gas

reservoir, in order to determine if the trends seen in the oilsimulations remained valid.

Finally, the effect of a change in aspect ratio wasinvestigated. Figure 7 illustrates this. Note that dimensionless

height, H D , is defined as follows;

H D =h f

H ..........................................................(4)

Basic Reservoir CharacteristicsIn order to model the effects of skin bypass fracturing, a

basic reservoir model was used, with the following

characteristics:-

Depth 5000 ftReservoir rock Sandstone

Fluid type Black oil or dry gas

Porosity 17 %

Well spacing 40 acres

Reservoir geometry Radial

Wellbore diameter 8.25 inchesNet Height, H 100 ft

Gross height 100 ft

Oil gravity 30º API

Gas gravity 0.65

Temperature 170º FReservoir pressure 2000 psi

All simulations were made using these basic parameters.

The VariablesThe following variables were used in the analysis;

Skin factor, S 0 to 25Fracture conductivity, F c 2000 to 5000 mdft (oil)

1000 to 2000 mdft (gas)

Permeability, k f 1 to 1000 md (oil)

0.01 to 10 md (gas)Aspect ratio, AR 0 to 2.5

Fracture height, h f 0 to 100 ft

The Theoretical Acid TreatmentThe main use for skin bypass fractures is as an alternative

to matrix acid treatments. Therefore, in order to show the

relative benefits of each treatment, two theoretical acid

treatments have been included as a comparison. These

treatments, which stimulate the entire zone equally, have theeffect of reducing the skin factor to zero. The effect of

reducing the skin factor, for the above reservoir, is as follows;

S = 25, J/J0 = 4.62

S = 5, J/J0 = 1.72

Figures 2 to 5 illustrate the relative effects of the skin

bypass fracs and the theoretical acid jobs.

It should be noted that in reality it would be very difficult,

although not impossible, to produce a skin factor in a

sandstone as low as zero after an acid treatment.

Limitations of Simulation ModelThe authors recognize that the methods used to simulate

the effects of the skin bypass fractures have a number of

limitations. In particular;

1. No vertical movement of fluids between the fractured

and unfractured sections. This quantity, which becomes more

significant as the H D decreases and the AR increases, controlsthe ability of the reservoir fluid to move up or down from the

unfractured sections into the fracture. If this movement was

simulated, with a discrete vertical permeability, it would allow



SPE 56473 SKIN BYPASS FRACS: PROOF THAT SIZE IS NOT IMPORTANT 5

easier access to the fracture for the reservoir fluids, thus

increasing the effectiveness of the fracture. Therefore, the lack of vertical permeability produces a pessimistic estimate of

production increase J / J 0 .

2. Single phase fluid flow. In order to simplify the

analysis and results, the authors decided to model fluid flow in

the reservoir and fracture as either black oil or dry gas.

3. Steady state flow. The simulation method did notmodel pseudo-steady state or transient flow.

All three of these limitations are areas in which further

investigation can and should be carried out. However, by

reducing the results to dimensionless quantities wherever

possible, it is believed that a many of the errors caused bythese limitations are minimised.

Additionally, although the absolute numerical values

produced are as accurate as possible given the circumstances,

the most significant results are the trends. These clearly

illustrate the effectiveness of skin bypass fracturing.

Results

The results from the simulation studies are contained inFigures 2 to 7.

Figures 2 and 3 show the variation of J / J 0 with proppant

volume for various combinations of F c , S and k f for the oil

reservoir. Also included on these graphs are lines indicatingthe production increase that would be obtained from the

theoretical acid treatments, which result in a uniform zero skin

over the entire interval. These results clearly indicate that only

relatively small quantities of proppant are required to produce

significantly better results than even the best acid treatments.

Figure 4 shows the same as above for the gas reservoir.

Although, the figure only shows k f = 1 md, the same trends

were seen over a wide range of k f .

It should be noted that the proppant volume is not only anindicator of treatment size, but also of treatment cost.

Generally speaking, larger treatments (i.e. greater volumes of

proppant) require larger fluid volumes, increased pumping

power and longer job times. This increased requirement for

equipment, personnel and materials results in greater costs.

Therefore, in order to minimize costs, proppant should be used

as efficiently as possible.

Figures 5 and 6 show the effect of AR on volume of proppant required to produce a required h f at a given F C .

Although the AR is usually not controllable, especially in a

skin bypass fracture which probably will not contact any

formation boundaries, it’s effects can be significant.

Figure 7 shows the effect on J / J 0 of variations in AR for

various values of H D. This graph clearly indicates that an AR

of 0.5 – radial frac geometry – produces optimum results. This

effect is so significant that under certain circumstances it may

be advantageous to limit the size of treatment, in order to

avoid hitting any boundaries, which could cause the fracture to

preferentially extend and the AR to significantly increase.

Figure 8 shows the relative quantities of proppant needed

for 20/40 Colorado Sand and 20/40 Carbolite, indicating that

the artificial proppant’s superior permeability means

substantial benefits in terms of proppant volume reduction,

which in turn means significantly reduced job costs.

The Theoretical Skin Bypass Frac

Prats6 introduced the concept of effective wellbore radius,

r w´ and dimensionless effective wellbore radius r wD´. After a

stimulation treatment, the well will produce at steady state asif from the new, effective radius;

J =k h

µ ln (r e /rw´) ..........................................(5)

Now, given that Prats defined dimensionless effectivewellbore radius, for low conductivity fractures, as;

r wD´ =k p w f

4 k f x f =

F CD

4 .........................(6)

and that dimensionless fracture length is defined as;

x fD = x f

r e .........................................................(7)

then equation (5) reduces to;

J =k h

µ ln[4/(F CD . x fD)] ................................(8)

With h f = H . Now, the pre-treatment, unfractured productivity

for a well with a skin factor S , is given by;

J 0 =k h

µ ln[r e /(r w . e-S )] ...................................(9)

Therefore;

J

J 0 =

ln[r e /(r w . e-S )]

ln[4/(F CD . x fD)] .................................(10)

This is a more specific version of the relationships derived by

McGuire and Sikora10, applicable to skin damaged wells with

low relative conductivity fractures.

Figures 9 and 10 show the effect this relationship has on

the oil reservoir used in the simulation studies. Figure 9 has S

= 5, whilst figure 10 has S = 25.

A more general form of this relationship, applicable to awide variety of reservoirs, is given by rearranging equation

(10) to give;

J

J 0 ln[r e /(r w. e-S )]

=1

ln[(4/ F CD . x fD)] ....(11)

This equation is plotted in Figure 11, which is the low

conductivity, skin damaged version of the McGuire-Sikoracurves.




Case Histories

Case #1 – IndonesiaThis well is located on the island of Sumatra, and was

treated in May 1998 with a skin-bypass fracture treatment.

Data on the well is as follows:-

Well type OilInitial production 16 bopd

Net height 33 ft

Permeability 8 md

Average depth 2774 ft TVD

Reservoir pressure 650 psiSkin factor (estimated) 15

The following treatment was pumped, using a 40lb organo-

borate crosslinked guar fracturing fluid:-

Pad volume 4,000 galsTotal gel volume (inc pad) 7,500 gals

Proppant volume 16,500 lbmMax proppant concentration 8 ppa

Pump rate 8 bpm

Following the treatment, the production of the well was asfollows:-

Initial oil rate 108 bopd

Sustained oil rate 89 bopd

Case #2 – South Texas1

This well was treated down heavy-wall coiled tubing.

Reservoir data is as follows:-

Well type Gas

Initial production 100 mscfd

Initial FWHP 70 psi

Depth 6,870 ft TVD

Net height (assumed) 13 ft

Reservoir pressure 3000 ft

Permeability 0.1 md

Skin factor (estimated) 11

The well was treated with a total of 14,000 lbm of 20/40

Econoprop, which was placed at a maximum rate of 9.6 bpm,

using a visco-elastic surfactant-based frac fluid. After the

treatment, the well was producing at the following conditions:-

Initial production 1 mmscfd

Sustained production 800 mscfd

Sustained FWHP 300 psi

This treatment illustrates that the technique of skin-bypass

fracturing is not necessarily limited to high permeability oil

reservoirs.

Case #3 – VenezuelaThis injection well was treated in August 1998 with a skin

bypass fracture treatment. The well consisted of two zones,

which were treated simultaneously. Reservoir data is as

follows:-

Well type Water injector

Initial injectivity 1,800 bwpdInitial FWHP 1,800 psi

Depth 6650 ft TVD

Net height 25 ft

Permeability 3000 md

Skin Factor > 100

The treatment consisted of 24,200 lbm of 20/40 Carbolite

proppant, pumped at 20 bpm, at a maximum concentration of

8 ppa. Overall fluid volume was 10,500 gals, of which 5100

gals was the pad. After the treatment the well performed as

follows;

Stabilized injectivity 4,000 bwpdStabilized FWHP 1,800 psi

Conclusions1. Significant and meaningful increases in production

can be obtained from relatively small propped hydraulic

fracture stimulation treatments.

2. Such treatments do not necessarily need to produce

propped hydraulic fractures with F CD values greater than 1,

provided the original skin factor is significantly positive.

3. Skin bypass fracture stimulation is a viable

alternative to matrix acidizing, especially in formations which

are unsuitable for acid treatments, for reasons such as poor

mineralogy.4. Skin bypass fracturing is more applicable, but not

limited to, higher permeability wells – this is because thesewells have higher intrinsic productive capacity, yet also tend

to have higher skins. Benefits are, thus, significantly better in

both relative and absolute terms in such wells

5. In many circumstances, skin bypass fracturing is

more cost effective than matrix acidizing.

6. In some situations a smaller fracture may produce abigger productivity increase than a larger fracture.

7. Further investigation should be carried out into this

subject. Areas for investigation include, multiphase flow,

pseudo-steady state and transient flow, vertical fluid flow and

the use and possible design of more sophisticated simulators.

Nomenclature AR = Aspect ratio

F c = Fracture conductivity, mdft

F CD = Dimensionless or relative conductivity

FWHP = Flowing well head pressure, psi

H, h = Net height

H D = Dimensionless fracture height

h f = Fracture height at wellbore, ft




J =Post treatment productivity index, bbls/day/psi

J 0 = Initial productivity index, bbls/day/psi

k f = Formation permeability, md

k p = Proppant permeability (in fracture), md

r e = Radial extent of reservoir, ft

r w = Wellbore radius, ft

r w´ = Effective wellbore radius, ft

r wD´ = Dimensionless effective wellbore radius, ftS = Skin factor

TVD = True vertical depth

V prop = Proppant Volume, lbm

w f = Fracture width, in

x f = Fracture half length, ft

x fD = Dimensionless fracture length

γ = Proppant Specific Gravity

φ π = Proppant porosity

AcknowledgementsThe authors would like to thank the following people for their

help in the preparation of this paper; Patricio Torres, BuddyShotton, Gino di Lullo, Grant Nevison and Dave “Primordial-Batch-Frac” Cramer for his help with the references.

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on Formation Grain Size – A Necessity in Fracturing Weakly

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10th Offshore South East Asia Confedrence & Exhibition,

Singapore, 6-9 Dec 1994

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Distributions Created by a Well with a Single HorizontalFracture, Partial Penetration or Restricted Entry”, SPEJ, Aug

1974, 347 – 360.

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Analysis for Fractured Wells”, JPT , July 1975, 887 – 892.

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Laplace Transformation to Flow Problems in Reservoirs”, 1949,

Trans., AIME, 186, 305-324.

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– Incompressible Fluid Case”, SPEJ, June 1961, 105-18, Trans..

AIME, 222

7. Moore, T.V.: “Definitions of Potential Productions of Wells

Without Open Flow Tests”, Bull., API, Dallas, 1930, 205.

8. Harder, M.L.: “Productivity Index”, API, Dallas, May 1936.

9. Elbel, J.L. and Sookprasong, P.A.: “The Use of Cummulative

Production Type Curves in Fracture Design”, SPEPE , Aug.1987, 191 – 198.

10. McGuire, W.J. and Sikora, V.J.: “The Effect of Vertical

Fractures on Well Productivity”, Trans., AIME (1960), 219, 401

– 403.

11. Nolte, K. G., and Economides, M. J.: Reservoir Stimulation, 2nd

ed., Prentice Hall, Englewood Cliffs, NJ, (1989)

12. Gidley , J.L., et al.: Recent Advances in Hydraulic Fracturing,

Monograph Series, SPE, Richardson, Texas (1989)

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SI Metric Conversion Factors

ºAPI141.5

131.5 + ºAPI * = SG

acre × 4.046 873 E+03 = m2

bbl × 1.589 873 E-01 = m3

bbls/d/psi × 2.305 916 E-02 = m3 /day/kPa

bopd × 1.589 873 E-01 = m3 /day

bwpd × 1.589 873 E-01 = m3 /day

bpm × 1.589 873 E-01 = m3 /minute

°F (°F - 32)/1.8* = °C

ft × 3.048* E-01 = m

gal × 3.785 412 E-03 = kg/m3

HHP × 7.457 0 E+02 = Watt

inch × 2.54* E+01 = mm

klbs × 4.535 924 E+02 = kg

lbm × 4.535 924 E-01 = kglbs/cu.ft × 1.601 846 E+01 = kg/m3

md × 9.869 233 E-04 = µm2

mdft × 3.008 142 E-04 = µm2.m

mscfd × 2.831 685 E+01 = sm3 /day

mmscfd × 2.831 685 E+04 = sm3 /day

ppa × 1.198 264 E+02 = kg/m3

psi × 6.894 757 E+00 = kPa

sq.ft × 9.290 304 E-02 = m2

*Conversion factor is exact.




Fig. 1 – Schematic Diagram Illustrating the Model Used for the Simulations

Fig. 2 – Graph Showing the Effect of Proppant Volume on Production Increase, J /J 0 , for a Fracture with an AR of 0.5 in a 100 mdOil Reservoir. Note that even a small fracture can match or exceed the results from a matrix acid treatment.

Unfractured

Unfractured

Fractured

x f

Permeability, k p

Width, w f

h f H = 100 ft

Skin Factor, S

Skin Factor, S

Permeability, k f

Permeability, k f

Permeability, k f

C o m m o n

D r a w d o w n

k f = 100 md, AR = 0.5, Oil

20/40 Colorado Sand

0

10

20

30

40

0 4 8 12 16 20

V prop , klbs

J / J 0

S = 25

S = 5Skin reduced from 25 to 0 by acid treatment

Skin reduced from 5 to 0 by acid treatment

5000 mdft

2000 mdft

5000 mdft

2000 mdft




Fig. 3 - Graph Showing the Effect of Proppant Volume on Production Increase, J /J 0 , for a Fracture with an AR of 0.5 in a 1000 md OilReservoir. Note that even a small fracture can match or exceed the results from a matrix acid treatment.

Fig. 4 - Graph Showing the Effect of Proppant Volume on Production Increase, J /J 0 , for a Fracture with an AR of 0.5 in a 1 md GasReservoir. Note that even a small fracture can match or exceed the results from a matrix acid treatment.

k f = 1000 md, AR = 0.5, Oil

20/40 Colorado Sand

0

2

4

6

8

10

0 4 8 12 16 20

Vprop, klbs

J / J 0



S = 25

S = 5

5000 mdft

2000 mdft

5000 mdft

2000 mdft

k f = 1 md, F c = 1000 mdft, AR = 0.5, Gas20/40 Colorado Sand

0

5

10

15

20

25

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

V prop , klbs

J /

J 0

S = 25

S = 5

Skin reduced from 25 to 0 by acid t reatment





Fig. 5 – Graph Showing Quantity of Proppant Required Against Fracture Half Length, x f , for Various Values of AR , for an F c of 2,000 mdft

Fig. 6 – Graph Showing Quantity of Proppant Required Against Fracture Half Length, x f , for Various Values of AR , for an F c of 5,000 mdft

F c = 2,000 mdft

20/40 Colorado Sand

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80

V prop , klbs

x f ,

f t

AR = 0.2AR = 0.5

(radial frac)AR = 2.0

F c = 5,000 mdft

20/40 Colorado Sand

0

20

40

60

80

100

120

0 40 80 120 160 200

V prop , klbs

x f ,

f t

AR = 0.2

AR = 0.5

(radial frac)AR = 2.0




Fig. 7 – Graph Showing the Effect of AR and H D on Productivity Increase J/J 0 , for a 1000 md, Oil Reservoir with S = 25. This effect is a highpermeability phenomenon and becomes less significant as permeability decreases.

k f = 1000 md, S = 25, F c = 5000 mdft, Oil,

0.0

2.0

4.0

6.0

8.0

10.0

0.0 0.5 1.0 1.5 2.0 2.5

AR

J / J 0

H D = 1.0

H D = 0.8

H D = 0.6

H D = 0.4

H D = 0.2

F c = 5000 mdft, AR = 0.5

0

4

8

12

16

20

0 5 10 15 20 25 30 35 40 45 50

x f , ft

V p r o p ,

k l b s

20/40 Colorado Sand

20/40 Carbolite Proppant

Fig. 8 – Graph Illustrating the Effects of Selection on Proppant Volume




Fig. 9 – Graph Showing the Theoretical Results for S = 25

Fig. 10 – Graph Showing the Theoretical Results for S = 5

S = 25

0.0

4.0

8.0

12.0

16.0

20.0

0.1 1 10

F CD

J / J 0

X f = 10 ft

X f = 20 ft

X f = 30 ft

X f = 40 ft

X f = 50 ft

S = 5

0.0

2.0

4.0

6.0

8.0

0.1 1 10

F CD

J / J 0

X f = 10 ft

X f = 20 ft

X f = 30 ft

X f = 40 ft

X f = 50 ft




Fig. 11 – Graph Showing the Theoretical Results for the General Case for Skin Bypass Fracs

General Case

0

0.1

0.2

0.3

0.4

0.5

0.01 0.10 1.00

F CD

x fD = 0.01

x fD = 0.05

x fD = 0.1

x fD = 0.2

x fD = 0.5

J

J 0 l n ( r e

/ r w

e - S )

Date post:	07-Jul-2018
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4.Paper for Chapter 5_SPE 56473 Skin Bypass Fracturing

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