Suspensions: Fundamentals and Applications in the Petroleum Industry

11 Suspensions in Hydraulic Fracturing

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Subhash N. Shah School of Petroleum and Geological Engineering, The University of Oklahoma, Norman, OK 73019-0628

Suspensions or slurries are widely used in well stimulation and hydraulic fracturing processes to enhance the production of oil and gas from the underground hydrocarbon-bearing formation. The success of these processes depends significantly upon having a thorough understanding of the behavior of suspensions used. Therefore, the characterization of suspensions under realistic con ditions, for their rheological and hydraulic properties, is very im portant. This chapter deals with the state-of-the-art hydraulic frac turing suspension technology. Specifically it deals with various types of suspensions used in well stimulation and fracturing processes, their rheological characterization and hydraulic properties, be havior of suspensions in horizontal wells, review of proppant set tling velocity and proppant transport in the fracture, and presently available measurement techniques for suspensions and their merits. Future industry needs for better understanding of the complex be havior of suspensions are aho addressed.

Τ Ή Ε H Y D R A U L I C F R A C T U R I N G T E C H N I Q U E for w e l l stimulation has been

used successfully since 1946 as a means of increasing o i l and gas pro duction. Since its inception, the science of hydraulic fracturing tech nology has advanced considerably. In the early 1950s, the pioneer operating and service companies contributed in the cooperative devel opment of well stimulation technique. In the 1970s, the U.S. government provided funds to develop tight gas sands and unconventional energy resources such as coal b e d methane. T h e most significant advance of the technology was made i n the 1980s when emphasis was placed on stimulating medium to high permeability reservoirs. T h e hydraulic frac turing technology has been successfully applied i n sand control when producing soft and unconsolidated formations. Treatment optimization and improving economics must become the focus of the industry as it 0065-2393/96/0251-0565$15.75/0 © 1996 American Chemical Society

In Suspensions: Fundamentals and Applications in the Petroleum Industry; Schramm, Laurier L.; Advances in Chemistry; American Chemical Society: Washington, DC, 1996.


566

SUSPENSIONS: F U N D A M E N T A L S & APPLICATIONS IN P E T R O L E U M INDUSTRY

moves into the future. In the past decade, exciting progress has been made in the development of new fluids and proppant, accurate downhole tools, and highly sophisticated interpretation techniques for monitoring hydraulic fracturing treatments (1-4). In the application of hydraulic stimulation process, a viscous fluid (generally a non-Newtonian fluid) is p u m p e d down into the w e l l under high pressure to initiate and extend induced fractures. After this stage, another viscous-fluid stage containing proppant (solid particles; we use " p r o p p a n t " and " p a r t i c l e " interchangeably) is injected down into the wellbore at high rate and pressure to maintain the fracture geometry created by the previous clean viscous-fluid stage. T h e other function of the proppant-laden fluid (suspension or slurry; we use "proppant-laden fluid," "suspension," and " s l u r r y " interchangeably) stage is to keep the created fracture open after p u m p i n g has ceased. Fluids containing proppant account for 2 0 - 8 0 % of the total volume of a fracturing treat ment (5). Therefore, it is imperative to understand the behavior of these suspensions. Fracture geometry and extension d u r i n g treatment depend largely on the rheological characteristics of the clean viscous fluid and suspen sions or slurries prepared w i t h viscous carrier fluids. Particle settling and distribution in the fracture also are affected significantly by sus pension properties. W e discuss the suspensions used in w e l l stimulation and hydraulic fracturing processes. T h e following sections pertain to various types of suspensions used in well stimulation and fracturing processes, their rheological characterization and hydraulic properties, behavior of sus pensions in horizontal wells, a state-of-the-art review of proppant settling velocity and proppant transport in the fracture, presently available mea surement techniques for suspensions and their merits, and, finally, a summary and conclusions on the use of suspensions i n w e l l stimulation. Future industry needs for better understanding of the complex behavior of suspensions are also mentioned in this section.

Types of Suspensions A fracturing fluid is used primarily to wedge open and extend a fracture hydraulically and to transport and distribute the proppant along the fracture. Success of a fracturing treatment, among other things, highly depends on selection of the proper fracturing fluid. Selection must be made based on known fluid and reservoir properties. F l u i d properties strongly govern fracture propagation and the proppant distribution and placement. Today, many different types of fracturing fluids are available for reservoirs, ranging from shallow low temperature formations to those that are deep and hot. Comprehensive details and their design infor-



11.

SHAH

Suspensions in Hydraulic Fracturing

567

mation are outside the scope of this chapter. F o r the treatment of a specific formation, one must consult published engineering and labo ratory test data available from various sources, such as service companies. Fracturing fluids are classified as Newtonian, polymer solutions, cross-linked polymer solutions, emulsions, micellar solutions, foams, and gelled-organic liquids in solution with a liquefied gas (4, 6-8). As men tioned previously, proppant or solids are often used to h o l d the fracture open after the fracturing treatment has been completed. The solid par ticles become lodged in the fracture, keeping it open and providing a channel for the oil and gas to flow more easily from the formation to the wellbore. Because of its low cost, silica sand is by far the most com monly used proppant today. M a n y manufactured materials, such as sin tered bauxite (aluminum oxide), alumina, cordierite, mullite, silicon carbide, and some ceramic oxides, are also used (9-12). Coated proppants, such as resin-coated sands, are also being used (13). T h e manu factured materials and coated proppants are significantly stronger than sand and are used in deep formations where high in situ stresses crush sand or where large proppant permeability is needed. Suspensions or slurries used for well stimulation are prepared by adding a known quantity of proppant to the carrier fluid of choice to obtain a desired proppant concentration. T h e proppant concentration is usually referred to as the amount of solids added in a gallon of carrier fluid. A 2 0 - 4 0 U . S . mesh size is the most commonly used proppant for hydraulic fracturing. H o w e v e r , other particle sizes such as 8 - 1 2 , 8 - 1 6 , 1 0 - 2 0 , 1 2 - 2 0 , and 4 0 - 6 0 mesh are also used. The carrier fluids are non-Newtonian in nature and provide high viscosity and excellent drag-reducing characteristics. These fluids include polymer solutions, cross-linked polymer solutions, emulsions, and foam fluids. Many types of water-soluble polymers such as guar gum, hydroxypropyl guar ( H P G ) , carboxymethyl hydroxy propyl guar, and hydroxyethyl cellulose can be added to water to make it more viscous to improve proppant transport properties. F o r stability at higher temperatures, metal ion cross-linkers such as borate, Ti(IV), Zr(IV), and Al(III) are also used. F o r water-sensitive formations, oil-based fluids such as gelled ker osene, diesel, distillates, and many crude oils are used. F o r better fluidloss behavior, polymer emulsions are also sometimes used. Polymer o i l in-water emulsions normally consist of 6 0 - 7 0 % liquid hydrocarbon as an internal phase and 3 0 - 4 0 % gelled water as the external phase. B e cause of their high solubility, alcohol-based fracturing fluids are used for treating gas-producing formations. Foams are primarily used in low pressure reservoirs to improve cleanup and minimize formation damage. Suspensions or slurries are prepared with all of these carrier fluids. Thus, it is very important to understand the behavior of these suspensions.


568



Rheological

Characterization

In the early days of fracturing treatments, proppant concentrations were 1-6 lb/gal, but now concentrations of 14 and 16 lb/gal are not uncom mon. The effect of proppant on fluid viscosity (under laminar flow in the fracture) at low solid concentrations is not significant but is at higher solid concentrations. W i t h recent advances i n placement of high p r o p pant concentrations and the desire to predict fracture geometry and extension accurately, it has become important to include the effect of proppant concentration on the viscosity of fracturing fluids i n the cur rently available fracture design simulators. Rheological characterization of neat fluid (i.e., fluid without prop pant) is relatively w e l l understood. H o w e v e r , the rheological charac terization of fracturing suspensions or slurries is not w e l l known. T h e primary reason for not determining the rheological properties of slurries is the difficulty of particle settling at low shear rates while making mea surements with a rotational-type viscometer, a pipe viscometer, or a slot-flow device. T h e proppant must be kept i n uniform suspension to obtain meaningful data. Particle shape, size, and density; polymer gelling-agent concentra tion; solids concentration; test temperature; and fracture shear rate affect viscosity increase that results from the addition of a solid in the fractur ing fluid. Gardner and Eikerts (14) used a large closed-loop pipe viscometer to evaluate the effect of 2 0 - 4 0 mesh sand on the viscosity of aluminatecross-linked 60-lb carboxymethyl hydroxyethylcellulose/1000 gal gel. They concluded that the effect of sand on gel viscosity was a larger increase in slurry viscosity in the fracture than predicted from Newtonian slurry models. E l y et al. (15) studied the effect of sand concentration on 40 lb H P G / 1000 gal fluid using a Brookfield viscometer and reported an increase in viscosity caused by the addition of sand. F o r 50 l b / 1 0 0 0 gal crosslinked H P G fluid containing 2, 4, and 6 lb/gal sand at 190°F, the power law flow behavior index, n , decreased whereas the consistency index, K , increased w i t h increasing sand concentration. F o r fracturing suspensions prepared with only linear H P G , Nolte (16) recommended the following modified L a n d e l et al.'s (17) equation for Newtonian suspensions. Mr = [1 - (Φ/Φ^)\-

25η

(1)

where μ is the relative viscosity and is the ratio of the suspension vis cosity to the carrier fluid viscosity at the same shear rate, φ is the volume fraction solids and is the maximum packing concentration. F o r u n i form spherical particles, 0 , the maximum packing concentration varies Γ

max

m a x



11.

SHAH

569


between 0.74 (body-centered cubic) and 0.52 (simple cubic lattice). Nolte simply modified the power o f —2.5 in L a n d e l et a l / s equation to —2.5n. T h e modified L a n d e l et a l / s equation 1 is independent o f shear rate. Results (16) of tests conducted in a specially designed cup and bob with 20, 30, and 4 0 l b H P G / 1 0 0 0 gal fluids containing 1 8 - 2 5 mesh neutrally buoyant beads at four volume fractions o f particles were re ported. It was claimed that for all polymer and particle loadings, rheo grams from 0 to 150 s~ essentially produced the same η value as the corresponding fluid without particles. T h e Κ values, however, were higher because of the particles. T h e results are compared w i t h L a n d e l et al.'s equation 1 and w i t h the following equation of F r a n k e l and A c r i vos (18). l

Mr =

9/8

(0/0max)

1/3

[I ~ (0/0max)

(2) 1/3

L a n d e l et a l / s equation appears to be reasonably good w i t h less viscous suspensions, whereas the F r a n k e l and Acrivos equation seems tofitthe data w i t h more viscous slurries and at higher particle concentrations. It is assumed that all of Nolte's data are taken at room temperature. K e c k et al. (19) studied the effect o f proppant on the effective vis cosity o f non-Newtonian fluids and presented the following modified Eiler's (20) expression that includes the effect of shear rate, temperature, gel concentration, and proppant concentration.

Mr =

1 + {0.75(e

15n

-

l

n

1000

i) -( - ^ e

}

1.25φ 1 - 1.5Φ

12

(3)

Tests were conducted w i t h a rotational viscometer with 30, 40, and 60 lb H P G / 1 0 0 0 gal fluids containing neutrally buoyant 6 0 - 1 0 0 mesh sty rène divinylbenzene beads at concentrations up to 12 lb/gal and tem peratures up to 65.5 °C. D a t a were gathered only at three shear rates: 5, 170, and 1000 s" . T h e i r modified E i l e r ' s equation was based o n correlating relative viscosity as a function of clean fluid η values, solids concentration, and fracture shear rate. T h e gel concentration and tem perature effects were incorporated into n. F i g u r e 1 depicts the effect of polymer concentration on the relative viscosity of suspension at a shear rate of 170 s and 23.9 °C. It can be seen that the lower polymer concentration has the greater viscosity ratio than the higher concentra tions and that the difference between these increases w i t h volume frac tion solids. B y using field and service company data, Jennings (21) developed the following two simple empirical equations, one for non-cross-linked 1

_ 1


570



4

0.1

0.3

0.2

0.4

Volume Fraction

Figure 1. Experimental viscosity ratio vs. volume fraction: polymer con centration effects. (Reproduced with permission from reference 19. Copyright 1992 Society of Petroleum Engineers.)

fluids (base gels) and one for cross-linked gels, to estimate relative vis cosity. These correlations are a function of sand concentration only and are independent of gelling agent concentration, temperature, and frac ture shear rate. Non-cross-linked fluids are defined as follows: = μ (0.21c + 1) Ά

(4)

where c is the sand concentration in lb/gal. Cross-linked fluids are defined as follows: Mam

= M (0.075c + 1) a

(5)

Shah (22, 23) investigated the rheological behavior of moderately concentrated (up to 35 vol%) suspensions prepared with non-Newtonian carrier fluids. Experimental data on the Poiseuille slit flow of suspensions of sand in H P G solutions were presented. F l o w data were gathered by pumping various non-Newtonian slurries into an 8-ft-tall vertical ap paratus, with care taken to avoid settling difficulties. T h e polymer solutions or base gels and suspensions exhibited pseu doplastic non-Newtonian behavior, and they were characterized by the following O s t w a l d - d e Waele or power law fluid model.


11.


SHAH

τ = K(i)

571

(6)

n

or


Ma =

(7)

(r/i) = K(y)"

Figure 2 presents an example of a logarithmic plot of the apparent viscosity versus slot shear rate data of 60 lb H P G / 1 0 0 0 gal fluid w i t h and without proppant at 60 °C. It can be seen that for the shear rate range studied, a linear relationship between apparent viscosity and shear rate on a logarithmic scale for all fluids w i t h and without sand is evident. This means that the rheological behavior of these fluids, H P G solutions as w e l l as slurries, can be adequately described by the power law expression given by equation 6. Further, a significant increase in ap parent viscosity of H P G solution is seen because of the addition of sand. T h e viscosity increase is more pronounced at lower shear rates than at higher shear rates. T h e n and K values are plotted as a function of volume fraction solids i n F i g u r e 3 for fluids tested at 60 °C. It can be seen i n this figure for these fluids, as the volume fraction solids increases, the n values decrease whereas the K values increase. This means the degree of the non-Newtonian character of the base gels increases by adding sand. T h e assumption that the n values of sand-laden fluids remain the same as the carrier fluid η by previous investigators does not seem to be valid. s

s

s

s

s

1,000

* BASE GEL 20/40 SAND Ο 2 PPG • 4 PPG Δ 6 PPG • 8 PPG

500

100

200

300

500

SLOT S H E A R RATE, 1/SEC

Figure 2. Viscosity vs. slot shear rate, 60-lb HPG slurries at 60 °C. (Re produced with permission from reference 22. Coypright 1993 Society of Petroleum Engineers.)



572


IF

1 0.1

0.0021 0

1 0.2

1 0.3

1 0.4

VOLUME FRACTION OF SOLIDS

Figure 3. K and n vs. volume fraction of solids at 60 °C. (Reproduced with permission from reference 22. Copyright 1993 Society of Petroleum Engineers.) s

s

T h e dramatic effect on n and K is seen b y increasing solids content. T h e n values, however, seem to level off or reach asymptotic values at higher solids concentration. Regression analysis was performed on n and K data of F i g u r e 3 by using the equations of the following form: s

s

s

s

s

n = a - (exp)fo

(8)

K = a- (exp)M

(9)

s

n

n

s

h

where a b , a , and b are constants. T h e equations 8 and 9 reduce to the H P G solution for zero volume fraction solids. T h e variations of con stants a and a and exponents b and b i n equations 8 and 9 as a function m

n

n

k

k

k

n

k


11.

SHAH


573

Temp °C * A 26.7 O · 43.3 • • 60.0


H 2.5 2 ï 1.5 5

6

Polymer Concentration, K g / m

3

Figure 4. Parameters a„ and b vs. polymer concentration. (Reproduced with permission from reference 23. Copyright 1993 Society of Petroleum Engineers.) n

of polymer concentration and temperature are presented in Figures 4 and 5. Figure 6 depicts the effect of volume fraction of solids on the relative viscosity for 40 lb H P G / 1 0 0 0 gal fluid at 100 s" and at test temperatures of 26.7, 43.3, and 60 °C (23). A Newtonian relative viscosity-concen tration curve based on Thomas (24) and M a r o n and Pierce (25) corre lations are shown for comparison. 1

0.12 0.1

6.5

0.08

0.04

*

A

Ο

·

•

•

26.7 43.3 60.0

0.02

4.5

Polymer Concentration, Kg/m

Figure 5. Parameters and b vs. polymer concentration. (Reproduced with permission from reference 23. Copyright 1993 Society of Petroleum Engineers.) k



574


0« 0.05

1

0.1

1

0.15

i

ι

0.2

0.25

0.3

ι

ι

I

0.35

0.4

0.45

VOLUME FRACTION OF SOLIDS

Figure 6. Effect of sand concentration on relative viscosity. HPG 4.8 kg/ m ; shear rate = 100 s' . (Reproduced with permission from reference 23. Copyright 1993 Society of Petroleum Engineers.) 3

1

It can be seen from F i g u r e 6 that the relative viscosity increases gradually at lower sand concentrations but increases very rapidly at higher sand concentrations. A t low particle concentrations, the hydrodynamic interaction is a dominating factor, whereas at higher particle concentrations, interparticle interaction becomes a dominating factor. Furthermore, in F i g u r e 6, relative viscosities of all non-Newtonian slur ries at all temperatures are substantially lower than those predicted by either Thomas equation or M a r o n - P i e r c e equation for Newtonian slur ries. Large errors may be committed by not using proper equations for non-Newtonian slurries. T h e effects of polymer concentration, shear rate, and temperature were also investigated. As the polymer concentration increases (power law exponent η decreases, that is, the viscosity of the carrier fluid or suspending liquid increases), the relative viscosity of the pseudoplastic slurry decreases. It is important to know that the more viscous the fluid (lower η value), its relative viscosity w i l l deviate further from the N e w tonian predictions. Relative viscosity decreases as the shear rate i n creases. A g a i n , the effect is more pronounced at higher solid concen tration than at low solid concentration. Temperature also has a dramatic effect on the relative viscosity of slurries. This effect is partly due to the reduction in carrier fluid viscosity because of thermal effects. The relative viscosity increases as the temperature increases. O v e r a l l , Shah's exper imental results agree more closely with Keek's than w i t h any other re ported study.


11.

SHAH


575

It should be remembered that unlike Nolte's proposed equation, Shah's results show definite shear rate-dependent relative viscosities. Also, power law exponent, n , of slurry is not the same as the carrier fluid η as assumed by N o l t e . P r u d ' h o m m e (26) has also reported de creasing η values with increasing volume fraction solids. s


Hydraulic

Properties

Hydraulic properties, that is, friction-loss calculations of proppant-laden fluids or slurries, are very important not only i n the design of any h y draulic fracturing treatment but also in real-time monitoring of fracturing treatments. Recent advances (27, 28) in real-time fracture analysis have necessitated an accurate knowledge of bottomhole treating pressure ( B H T P ) . T o estimate B H T P , an accurate prediction of friction pressures of fluids in the flow conduit is required. It is possible to obtain the B H T P from the surface pressure with the following equation: Pwt = Pwh + Ph ~ Pf

(10)

The wellhead pressure, p , can be measured with good accuracy. The hydrostatic pressure, p , can be calculated accurately w i t h recent advances i n radioactive densimeters. Methods for predicting friction pressures of clean fracturing fluids have been w e l l established. T h e fric tion pressures of gelled fluids through tubular goods, pf, are more difficult to predict accurately when they are cross-linked or contain proppants. Several studies have been reported to determine friction losses in turbulent flow of slurries. Hannah et al. (29) presented an approach i n which they compared expressions for the friction pressure of the slurry and clean fluid. In their analysis, they assumed Blasius'(30) turbulent Fanning friction factor versus Reynolds number equation for Newtonian fluids. The following expression for estimating slurry friction pressure knowing the clean fluid friction pressure is proposed. w h

h

P, = P n W ( p ) 02

r

08

(ID

where μ is the relative viscosity (ratio of slurry apparent viscosity to fluid apparent viscosity) and p is the relative density (ratio of slurry density to fluid density). A s is evident from the data shown i n F i g u r e 7, despite some data scatter, the correlation seems to work reasonably w e l l for H P G fluids and for sand concentration up to 7.5 lb/gal. Shah and L e e (31) presented an approach for predicting friction pressures of fracturing slurries that is based on an analytical method and uses nondimensional quantities. F r o m flow data for various slurries i n multiple pipes, generalized correlations were developed and presented that incorporated variables such as particle size, particle density, particle Γ

r


576


ι ι ι ι Iι ι ι ι Iι ι

ι I

1 1 1

C r o s s - L i n k e d

HPG

1

Ο

60

-

Ο

5 0

l b


HPG

-

0

4 0

l b


HPG

Δ

3 0

lb

C r o s s - L

t nked

HPG

l b

t

1

"

i

l

1111

l

M

M

Δ A


/

Δ

-

c

0

0

/ ft • /0 0

-

-

V

0

-

0 1

-

I

• 1

I

1 1 1

1 1

2

i

1 1 1 1 3

PROPPANT

M

I

4

L

I

M

•

1

5

6

CONCENTRATION

(

L - > - i -

1

1 1 1

M

i

l

7

lb'gal )

Figure 7. Correlation for proppant friction correction factor—5.5 in. csg, 1.9-in. tubing annulus. (Reproduced with permission from reference 29. Copyright 1983 Society of Petroleum Engineers.) concentration, fluid density, flow rate, polymer gel concentration, and pipe size. F o l l o w i n g Molerus and Wellmann's (32) analytical approach, the friction pressure o f slurry, p , is expressed as the sum o f friction pressure of clean fluid, pa, and an additional friction pressure, p , caused by par ticle present i n the fluid. T h e experimental data are correlated w i t h the expression of the form t

p

(vjv)

= v

soo

(12)

+ (N )~ * Sl

Frp

s

where ν is the average fluid velocity, v is the mean slip velocity between particle and fluid, v is an infinite slip velocity parameter, N is particle F r o u d e number, and s and s are empirical parameters determined from experimental data. T h e clean fluid friction pressure can be estimated w i t h either the recently published correlations and method described (33) or the aid of s

soo

F r p

Y

2


11.


SHAH

577

other similar methods. T h e additional friction pressure resulting from the presence of proppant can be estimated with the aid of the diagram presented in F i g u r e 8. F r o m this figure, knowing N and N * supplies the value of (v /v). W i t h this value, the dimensionless pressure drop, pa, is calculated as follows: F r p

F r

s


Pd

(13)

1 - (vjv)

T h e additional pressure drop, p , is then calculated w i t h the following equation: p

gy(pp ~ Pf)Lg

(14)

where c is the fractional volumetric concentration of proppants and is defined as v

(p ~ s

Pf)

P

Pf)

_ (p -

(15)

T h e friction pressure of slurry, p , is then simply a summation of ρπ and pp. T h e percent friction pressure increase over base gel was found to be a strong function of sand concentration and flow rate. A t a constant t

10"

— > ΙΟ"

Experimental Region Extrapolated Region

3

> d

£ 10" Φ c ο

4

(Λ

k ιο·

!

1X10-

10

Η

0

4

20 40 60 80 100 120 140 160 180 200 220 240 Particle Froude Number, Fr p

Figure 8. Master curve. (Reproduced with permission from reference 31. Copyright 1986 Society of Petroleum Engineers.)



578


flow rate, the percent friction pressure correction increases as sand con centration increases. A t a given sand concentration, however, the cor rection decreases as rate increases. A t the same flow rate, percent cor rection for more viscous slurry is lower than that for less viscous slurry. Furthermore, slurry shows higher percent correction w h e n it is p u m p e d down the annulus than tubing. Also, as particle size increases, the percent friction pressure correction increases significantly. A t lower proppant concentrations, the effect of particle density is minimal, but at higher proppant concentrations, this effect may be important. F i g u r e 9 shows a plot of pressure drop versus flow rate for fluids containing 40 lb H P G / 1 0 0 0 gal w i t h and without sand in a 2 / - i n fieldsized tubing. A n excellent agreement is seen between measured pres sures and laboratory predicted pressures. Figure 10 depicts a comparison between predicted percent friction pressure increase over base gel and actual field data acquired d u r i n g a fracturing treatment. A reasonably good agreement between the two is seen. Based on Bowen's (34) procedure, L o r d and M c G o w e n (35) devel oped a statistical based correlation of drag ratio versus fluid velocity. 7

4

6 8 10 12 14 16 Flow Rate, b b l / m i n .

18

8

20

Figure 9. Comparison of actual friction pressures at various flow rates for 40 lb HPG/1000 gal and sand-hden 40 lb HPG/1000 gal fluids in 2 / -in. tubing. (Reproduced with permission from reference 31. Copyright 1986 Society of Petroleum Engineers.) 7

8


11.

579

SlfAH Susvensions in Hydraulic Fracturing 110 100 90 ο

(Λ

S υ c

Φ _


Ο Δ

Actual Field Data Predicted from Present Study

80

10

3

Data Points

®

70

5) Ο

g 8 60

*S Is |S it c 8

5 0

Δ ΔΑ

Ο

40 30

Φ

Q.

20 10 0 2 4 6 8 10 12 14 20/40 Mesh Sand Concentration, lb/gal

Figure 10. Comparison of field data with predictions. (Reproduced with permission from reference 31. Copyright 1986 Society of Petroleum Engi neers.) T h e laboratory data of Shah and L e e (31) with various H P G solutions and slurries were used for this purpose. T h e i r initial correlation given by equation 16 below was shown to considerably overpredict friction loss of dense slurries during injections down long vertical tubing strings. In ( l / σ ) = 2.1505 - 8.024/t) - 0.2365 G/V - 0.1639 In G - 0 . 0 5 2 6 6 p - e

(16)

1/G

where σ is the drag ratio and is defined as the ratio of frictional pressure drop of gel or slurry and water, ν is the average fluid velocity i n ft/s, G is the H P G gel concentration in l b / 1 0 0 0 gal, and ρ is the proppant con centration i n lb/gal. T h e authors speculated that some heterogeneous flow phenomena, such as particle migration to the centerline, could possibly be responsible for the observation of lower than expected friction loss values during the treatment. Figure 11 depicts a comparison of laboratory data to correlation predictions of drag ratio versus velocity. It can be seen that the corre lation developed is diameter invariant. T h e laboratory correlation was


580


Pipe Diameter, inch •

0.824 1.038 1.367 Eq.(9)

Ο Δ ο

α

in _φ

ο

c g


*0.95) and therefore its inclusion, especially w h e n raised to a fractional power, has essentially no influence on correlation predictions. T h e i r equation appears i n the following form:


D

s

V

d

V g ^ ( F - 1) p

s

= 1.85c *

0 1 5 3 6

(l

-c) ^ 0

6 4 (

X(dp/d)-

'N

0378

2

6

)

'F -

,om

Rep

0

30

T h e data were extracted from a number of experimental investigations reported in the literature for the development of the previous equation. T h e O r o s k a r - T u r i a n correlation and others appearing i n the literature were all developed to describe critical deposition velocity of Newtonian carrier fluids w i t h various solid types, sizes, and concentrations. Shah and L o r d (49, 50) generalized the O r o s k a r - T u r i a n correlation to increase its capability to correlate critical velocity measurements with non-Newtonian carrier fluids. T h e parameter F was eliminated from the generalization because of its insignificant contribution to the correlation results and because it would be undefined for the laminar flows associated w i t h many of the non-Newtonian fluid measurements. T h e generalized form of equation 26, w h i c h can be applied to either critical deposition or resuspension velocity, is as follows: V

d

Vgdp(F. - 1)

= Ye°

1 5 3 6

(1 - c )

0 3 5 6 4

X (d /d)~ ·NL ·F p

w

Z

030

(27)

where coefficient Y and exponents w and ζ are adjustable constants that can b e evaluated by regression analysis for particular critical velocity data sets. Apparent viscosity, μ , is substituted into the modified Reynolds number expression to NR further generalize equation 26 to n o n - N e w tonian fluids. T h e rheological properties and model parameters o f fluids tested are listed in Table I. &

C


11.

SHAH

Suspensions in Hydraulic Fracturing T a b l e I.

Rheological Properties a n d M o d e l Parameters of F l u i d s Tested at 26.7 °C

Fluid

η

1.2 k g / m H P G 2.4 k g / m H P G 4.8 k g m H P G 7.2 k g / m H P G 4.8 k g / m cross-linked HPG 7.2 k g / m cross-linked HPG P o l y m e r emulsion 3

3

3

3

587

K(Pa-s ) n

y

w

ζ

0.941 0.719 0.486 0.377

0.00188 0.0546 0.6942 2.7195

3.4260 2.2573 0.1743 0.2360

0.6696 0.4691 0.4830 0.4513

-0.1481 0.0294 0.5276 0.4274

0.446

0.9719

0.0639

0.3786

0.7858

0.304 0.516

4.3091 2.9733

0.0613 0.2075

0.4356 -0.067

0.6476 1.0297


3

3

SOURCE: Reproduced with permission from reference 23. Copyright 1993 American In stitute of Chemical Engineers.

F i g u r e 17 shows critical deposition velocities for various fluids con taining 6 lb/gal 2 0 - 4 0 mesh sand in various field-size tubular goods. Note that critical velocities are very dependent on pipe size. M u c h greater velocities are required for the larger pipes to minimize particle settling. F u r t h e r m o r e , the critical deposition velocities of all fracturing slurries are substantially lower than those of sand-water slurries, and the critical deposition velocities of cross-linked slurries are substantially lower than those of base fluid slurries. Slightly higher critical resuspension velocities than critical depositional velocities for all fluids were reported. H i g h e r critical deposition and resuspension velocities are also re quired for proppant that is denser than sand. F o r less viscous fluids,

Figure 17. Effect of pipe diameter on critical deposition velocity (based on model predictions). (Reproduced with permission from reference 49. Copyright 1990 Society of Petroleum Engineers.)


588


both velocities increase slightly with increasing proppant concentration; however, they are independent of proppant concentration for more vis cous fluids.


Proppant

Transport

O n e of the most important factors in the effectiveness of the hydraulic fracturing treatment is the ability to predict the settling velocity of proppant under fracture conditions. T h e transport of proppant and the final distribution of proppant in the fracture highly depends on the ac curate estimation of settling velocity of proppant. T h e length of the propped fracture, the conductivity of the propped fracture, and height of the propped fracture are consequently affected by the settling velocity. In the last decade, several investigators have addressed the subject of particle motion in fracturing fluids. Experiments to determine particle settling velocities under dynamic conditions involved large vertical fracture flow models (51-54), flow loops w i t h single particles suspended in a vertical column (55), and rotating concentric cylinder devices in which particle fall rates could be observed (56-60). A detailed discussion regarding correlations developed from the data obtained from these apparatuses is beyond the scope of this chapter. A comprehensive list of available correlations has been presented by Daneshy (61). T h e fol lowing are, however, some of the correlations presently available that one can use to estimate single particle settling velocity i n fracturing fluids: Novotny (60) C

D

24 = T ^

N

R e p

Suspensions: Fundamentals and Applications in the Petroleum Industry

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