Particle Transport Through Perforations

C. Gruesbeck, * SPE, Exxon Production Research CO.

R.E. Collins, ** SPE, U. of Houston

Abstract This paper describes a new procedure for determining particle-transport efficiency through perforations. Transport efficiency is the mass fraction of particles that are transported through the perforations relative to the total mass of particles injected. Our conclusions are based on laboratory studies of the transport of solid par­ticles carried through perforations by different fluids having a wide range of physical properties. Prediction of particle-transport efficiencies based on theoretical con­siderations described in this paper gives an accurate representation of experimental results.

Introduction Transport of solid suspensions through perforations in a pipe wall presents special flow problems during gravel prepacking and hydraulic fracturing operations. Both of these operations commonly are practiced in the field, yet surprisingly little experimental or analytical work is available to guide engineers in selecting operating condi­tions that ensure that the solid particles are transported efficiently through the perforations and into the formation.

Torrest and Savage 1 and Haynes and Gray 2 studied the transport of sand slurries in a casing with perfora­tions. In these studies, however, no attempts were made to assess either the effects of fluid properties or the number of perforations on particle transport. Further­more, no attempts have been made to establish a theoretical basis for estimating particle-transport efficiency.

This paper describes a new procedure for determining particle-transport efficiency through perforations. Transport efficiency is the mass fraction of particles that is transported through the perforations relative to the

• Now a medical student. **Now with U. of Texas.

0197-7520/82/0012-7006$00.25 Copyright 1982 Society of Petroleum Engineers of AIME

DECEMBER 1982

total mass of the particles injected. The design procedure described in this paper can be used (I) to ensure that par­ticles are transported through each perforation that ac­cepts fluid, (2) to ensure that a high-density pack is placed outside the perforated casing during gravel prepacking operations, and (3) to prevent bridging of proppant particles in the perforation tunnels during hydraulic fracturing treatments. The conclusions of this work are based on laboratory studies of the transport of solid particles carried through perforations by different fluids having a wide range of physical properties. Predic­tions of particle transport efficiencies based on theoretical considerations described in this paper give an ·accurate representation of experimental results.

This paper is divided into two main sections. The first section discusses experiments performed to measure the transport efficiency of particles through perforations in casing. The second section describes a particle-transport theory that can be used to extend the laboratory results to full-scale field systems.

Measurement of Transport Efficiency Particle-Transport Apparatus The dynamics of particle transport through perforations were studied with two transparent lucite well bore models. Both a 3-in.-ID and a 7-in.-ID lucite casing were fitted with perforations as shown in Fig. 1. The IO-ft-Iong sections had a 3-ft-long flow-development section, a 4-ft-long perforated zone, and a 3-ft-long rat­hole section. A maximum of 30 perforations could be us­ed in an experiment by inserting small lucite perforation tunnels in the casing wall. The diameter of the perfora­tion tunnels could be changed from III in. through V2 in.

Fig. 2 shows the auxiliary equipment used during these tests. We started each experiment by adjusting the slurry pump and the outlet valve to deliver the desired flow rate to the model and by turning on the stirrer to

857

Page 1 of 9 Halliburton Energy Services, Inc.Exhibit 1013

SLURRY FEED , flOW OEVELOPMENT

SECTION

PERFORATION TUNNEl OETAIL

PERFORATEO ZONE

I I2J

4' I2J

!~ I2J

121

! I2J

r ~J I RATHOLE SECTION \

SLURRY RETURN

Fig. 1-Transparent wellbore model.

I SAMPLING

POINTS

,...-'r-'-'--'"

J

.1

~

Fig. 2-Flow diagram of particle-transport apparatus.

disperse the particles in the fluid. The slurry entered the model through the inlet port and descended through the flow development section, which ensured that the par­ticles were distributed unifonnly in the casing before they entered the perforations. The flow rate and particle concentration entering each perforation were measured through sampling ports. We were careful to ensure that the equilibrium was not disturbed during sampling-i.e., the sampling lines had the same length as the return lines, and the amount of slurry removed during anyone experiment was less than 5 % of the total amount of slurry in the system. At the end of each experiment, we detennined the particle concentration by oven-drying and then weighing the recovered particles.

Summary of Systems Studied The fluids used in the particle-transport studies were brines with a density range of 62.4 to 109 Ibm/cu ft.

858

Uu - c """"

1.0 r--------------------,

0.8

0.6

0.4

I

0.2

(11 20/40 US MESH GRAVEL. 10 cp HEC WATER SOLUTION

(21 20/40 US MESH GRAVEL. TAP WATER

(31 10120 US MESH GRAVEL. TAP WATER

PARTICLE CONCENTRATION, 1 LB/GAL (0.043 VOL. FRACTIONI

• (11

\2) (31

o 3/8-INCH PERFORATION

• 1/2-INCH PERFORATION

0.0 ""--__ ~ ___ -L-___ L-__ ____'__ ___ ...J

0.0 0.2 0.4 0.6

FRACTION OF FLUID THROUGH PERFORATION. (q, ) q"

1.0

Fig. 3-Particle-transport efficiency through one perforation in 3-in. casing at 0.055 cu ft/sec.

> U Z w

Q u. u. w >- Uu a: ~

-a~ Vl z " a: >-w -' u i= a:

" 0.

1.00-------------------,

0.8

0.6

! I

I 0.4

0.2

III 20/40 US MESH GRAVEL. 40 cp HEC WATER SOLUTION

(21 20/40 US MESH GRAVEL. 86.1 LB/FT3 CaCI2 (4.2 cpl

(31 15/18 US MESH BAUXITE (232 LB/FT31. TAP WATER

(11

(31

o 0.4 LB/GAL (0.0178 VOL. FRACTIONI

• 4.0 LB/GAL (0154 VOL. FRACTION)

0.2 0.4 0.6 0.8 1.0

FRACTION OF FLUID THROUGH PERFORATION.

Fig. 4-Particle-transport efficiency through one perforation in 3-in. casing at 0.133 cu ft/sec.

Hydroxyethyl cellulose was used to increase the viscosi­ty. The apparent viscosity of these fluids, measured at a rate of 500 seconds - 1 at 75 0 F, ranged from 1 to 100 cp. The particles used ranged in density from 162 to 230 Ibm/cu ft and included Brady sand, Ottawa sand, glass beads, and sintered bauxite. Five particle sizes were used: 40/60, 20/40, 15/18, 10/20, and 6/9 U.S. mesh. The particle concentration varied from 0.25 to 30 Ibm/gal of fluid (0.011 to 0.58 volume fractran).

Experimental Results The efficiency of particle transport through perforations was studied as a function of the properties of the par­ticles, the particle concentration, the fluid, the volumetric flow rate, the diameter and number of per­forations, and the casing diameter. Transport efficiency (E) was detennined from the mass fraction of particles that entered the perforations.

SOCIETY OF PETROLEUM ENGINEERS JOURNAL Page 2 of 9 Halliburton Energy Services, Inc.

Exhibit 1013

qci = 0.0667 FT3/SEC

Ci= 1 LB/GAL (0.011 VOl. FRACTION)

(UNIFORM FLUID FLOW THROUGH PERFORATIONS, 3 - INCH CASING)

1.8

40/60 US MESH

0.8

0.9

1.3

10 cp HEC SOLUTION

10/20 US MESH

0.92

0.94

1.14

Fig. 5-Gravel concentrations in multiple perforations.

n[

~ qi C E=----'----i=----'I __

qciCi . . . . . . . . . . . . . . . . . . . . . . . . . . . . (I)

Figs. 3 and 4 are typical of results obtained when the casing had only one perforation. The particle-transport efficiency, E, is shown for various particlelfluid com­binations. Note that the curves pass through the origin and that the transport efficiency is not affected by chang­ing perforation diameter from o/s to V2 in. (Fig. 3). Fur­ther, the transport efficiency is independent of initial par­ticle concentrations over the range of 0.4 to 4 Ibm/gal [0.0178 to 0.154 volume fraction (Fig. 4)].

When transport efficiency is plotted against the frac­tion of fluid rate through the perforation, as in Figs. 3 and 4, the slope of the transport -efficiency curve is equal to the ratio of the particle concentration in the perforation to the initial particle concentration in the casing (CIC i ).

The slopes of the efficiency curves for 20/40 U. S. mesh gravel transported with either a IO-cp or a 40-cp hy­droxyethyl cellulose solution are nearly 1.0 (Curve I in both figures). This means that under these conditions, the particles are transported into the perforation with negligible gravity or inertial separation. By contrast, when particles are transported by tap water, the concen­tration of the particles in the perforation is significantly lower than the initial slurry concentration (Curves 2 and 3 in both figures). This process is illustrated further by the data in Fig. 5. Here the ratio of the particle concen­tration, CIC i , is shown for experiments conducted in a casing with three perforations. The increase of CIC i is more pronounced for the larger gravel particles. Note that increasing the fluid viscosity from I to 10 cp results in a nearly uniform distribution of particle concentration into the perforations (i.e., CIC i = I).

A nearly uniform distribution of particles into the per­forations is desirable because it prevents buildup of high concentration in the lower part of the casing. If this is not done and if the particle concentration becomes suffi-

DECEMBER 1982

0 10

a: a: w

f- 8 w w f- :;< w <t :;< <t Ci Ci w

...J 6

Z u Q f= f- a: <t <t 4 a: 0..

0 W U. Cl a: <t w a: 0.. w

> 2 <t

0 0

MAXIMUM PARTICLE

CONCENTRATION - VOLUMEIVOLUME

0.08 0.15 0.21 0.27 0.31 0.58

• TAP WATER

• 100 cp HYDROXY ETHYL

CELLULOSE SOLUTION

2 4 6 8 10

MAXIMUM GRAVEL CONTENT - LB/GAL

Fig. 6-Bridging of particles in perforations.

30

ciently large, particles can bridge in the entrance of per­fomtions. This can result in incomplete packing of the region immediately outside the casing .

Bridging of Particles in Perforations The maximum particle concentration that can be transported through perforations is shown by the data in Fig. 6. Bridging inside a perforation occurs even at low gravel concentrations of 1/2 to I Ibm/gal (0.022 to 0.043 volume fraction) if the perforation diameter is two to three times the average particle diameter. Note, however, that when the perforation diameter is greater than six times the diameter of the average particle, bridg­ing does not occur even at gravel concentrations of 30 Ibm/gal (0.58 volume fraction). The ratio of perforation diameter to average particle diameter at which bridging occurs is insensitive to the viscosity of the carrier fluid; the ratios were virtually the same when either tap water or IOO-cp hydroxyethyl celulose solution was used. However, the size of the particle node that forms in the casing around a bridged perforation increases with in­creasing viscosity. If particle nodes form in opposite sides of the casing and if these nodes are sufficiently large, a particle bridge can form across the casing and prevent particles from being transported downstream of the nodes. Fig. 7 shows a particle bridge in a perforation and a typical node formed in the casing around a bridged perforation.

Discussion of Results Two conditions were observed in particle transport through perforated casing. At the start of injection, some particles bypass the perforations and fall into the rat hole, forming a bed. As injection continues, the height of the particle bed increases until some equilibrium height is established. This equilibrium bed height, which might be above the bottom perforation, depends on the total volumetric flow rate. If a flowing system is operating at equilibrium conditions and if the total volumetric flow rate increases, the equilibrium bed

859 Page 3 of 9 Halliburton Energy Services, Inc.

Exhibit 1013

ClOSE·UP OF BRIOGEO PERFORATION

Fig. 7-Particle node formed opposite a bridged perforation.

height will decrease until a new equilibrium state is established. If the total volumetric flow rate is increased sufficiently, the top of the equilibrium particle bed may be slightly below the bottom peltoration .

The final amount of particles transported through the perforations ultimately depends on two factors: (I) the presence of a void space outside the casing that can ac­cept the particles transported through the perforation, and (2) the size of the perforation relative to the size of the particle. Slurries containing more than 30 Ibm of gravel/gal of fluid (0.58 volume fraction) can be transported through perforations if the perforation diameter is sufficiently large.

Particle-transport efficiency can be found for a casing with any number of perforations by evaluating the forces the fluid exerts on the individual solid particles and by relating these forces to the trajectory that each particle follows in the perforated casing . Finding transport effi­ciency is complicated by the effect of velocity gradients on the rotation of particles and the effect of turbulent fluctuations on the dynamics of a freely moving body. These effects are poorly understood. Nevertheless, three forces have the greatest influence on the particle­transport efficiency: (I) gravitational forces causing the particles to settle , (2) viscous drag caused by the local relative velocity between the particles and the fluid , and (3) inertial forces , which occur because the particles resist changes of velocity .

Dimensional analysis shows that these forces can be expressed in two dimensionless groups.

(P-P.I) grc NFr= -2· ·· ·· · .. · . . . .. .. · .. · .. · .. (2)

Ps Vj

and

P rc ND =--KD(NRe ) . . . ..... . ... . . . .. .. . . .. (3)

P s d

These are the Froude number, which is the ratio of gravi­ty forces to inertial forces (Eq . 2), and the drag number,

860

z EQUAL AREAS z

RADIUS = rc RADIUS = rM

(A) CYLINDRICAL SEGMENT (B) CYLINDRICAL CASING

Fig. B-Cylindrical segment with perforations along the vertex.

which is the ratio of drag forces to inertial forces (Eq. 3). In addition to the forces expressed by these dimen­sionless groups , particle-transport efficiency also depends on the number and spacing of perforations, the depth of the rat hole , the flow distributions of the fluid through the perforations. and the relative dimensions of the particles to the perforation .

The two basic steps needed to calculate the transport efficiency are (I) to establish theoretical or experimental flow patterns or velocity distributions of the fluid in a perforated casing, and (2) to determine the trajectories of the particles by solving the equations of motion.

In Step I , the actual velocity distribution in a per­forated casing was approximated by a simple model based on ideal flow in a casing having the form of a cylindlical segment with perforations along the vertex (see Fig. 8). The motivation for this is that the flow problem is reduced from three dimensions (3D) to two (20) because the velocity field in the cylindrical segment is independent of the azimuthal angle. This axial sym­metry permits the use of any segmental angle, and for convenience we used the angle 7r. Using this approx­imate velocity field. we then determined the critical tra­jectory for a particle. The critical trajectory can be defin­ed as the path outside of which a particle will not enter a perforation ; a particle on the critical trajectory will just enter (not miss) the perforation . This simple model gave a surprisingly accurate representation of experimental data.

Prediction of Transport Efficiency The transport efficiency of particles transferred through perforations can be calculated if the critical trajectory of a particle just entering the bottom perforation can be deternlined. The importance of this critical, or limiting, trajectory is illustrated in Fig . 9 . As this figure shows, all particles inside the envelope (or surface of revolution) generated by this trajectory will enter the perforations , while all particles outside will sink in the rat hole. The particle-transport efficiency thus can be defined as the ratio of the mass of particles per unit time entering the perforations to the mass of particles introduced into the

SOCIETY OF PETROLEUM ENGINEERS JOURNAL Page 4 of 9 Halliburton Energy Services, Inc.

Exhibit 1013

casing per unit time. Since all the particles contained within the limiting trajectory of the bottom perforation will enter the perforations, we can see from Fig. 8 that the expression for the particle-transport efficiency reduces to a ratio of radii squared.

Fluid Velocity Field

One of the principal reasons for the apparent neglect of an analytical approach to particle transport has been a lack of quantitative data on flow patterns and velocity distributions for fluid flow in perforated casing. The fluid velocity can be obtained at any position in the cas­ing by solving the Navier-Stokes equations by numerical procedures. A good approximation to the flow patterns frequently can be obtained by solving the Navier-Stokes equations for potential flow-that is, with the assump­tion that the fluid is ideal (p =constant; fl =0) and that the flow is irrotational 07 x i7=0). These assumptions resuit in an approximate velocity field that is valid at high Reynolds numbers, except in a thin, viscous boun­dary layer.

Ideal fluid flow patterns and velocity distributions in a cylindrical segment, as idealized in Fig. 8, can be deter­mined by solving the Laplace equiltion for the velocity potential, <1>.

................... (5)

With appropriate boundary conditions, the radial and ax­ial components of the fluid velocity then can be deter­mined from

a <I> V r =-;;; ............................... (6)

and

a <I> V-=-- . ............................... (7) , az

For the case of one perforation with less than 100 % fluid removal, the perforation creates only a small per­turbation on the velocity distribution, except in the im­mediate vicinity of the perforation. For this reason, we can use an approximate solution that neglects pipe boun­daries-namely, a point sink plus unifonn flow in the semi-infinite domain.

qi <I>(r,Z) = ----+vjZ- .................. (8)

47r.J r2 + Z2

With multiple perforations, the pipe wall becomes a significant factor and must be considered. An analytical solution has been constructed for this case. This solution is suitable for any number of colinear perforations hav-

DECEMBER 1982

ENVELOPE OF LIMITING TRAJECTORY

'\ __ rM Z = 00

0 I ~ 0

~ z

//; PERFORATIONS zj Vr

'~I (0,0) ~ (0,0)

en ...J X ...J en <! <!

X s: >-<! '-' cc

I->- Z UJ cc Ci5 ::;: I- <! ::;: UJ U ::;: >-::;: en >-en

Fig. 9-Cylindrical coordinate system for particle trajectory calculations.

ing any distribution of flow rates and any depth of rat hole. This solution for n I colinear perforations treated as point sinks at distances of Zi, i = 1,2 ... n I' above the bottom of the rat hole and having flow rates of q i, i = 1,2 ... n I' is constructed for the approximate cylin­drical segment geometry indicated in Figs. 8 and 9, as

<I>(r,Z) =

i=1 j=1

where J 0 and J I denote the Bessel functions of the first kind of orders zero and one, respectively, and the (3/s are the roots of the equation

J I «(3/M)=O . ............................ (10)

This solution is in fact a special case of the Green's func­tion for the Laplace equation in a cylinder of infinite length and can be derived by using integral transfonn techniques, as, for example, in the text of Morse and Feshbach. 3

Differential Equations Describing Particle Motion If the flow field in a casing with colinear perforations is approximated by an axially symmetric system of point sinks on the axis of a cylindrical segment as in Figs. 8 and 9, the particle trajectories are the same in all meri­dian planes as shown in Fig. 9. The problem then becomes 2D, and distances can be resolved in axial and radial components in the cylindrical coordinate system.

861 Page 5 of 9 Halliburton Energy Services, Inc.

Exhibit 1013

Consequently, the solution of only two simultaneous equations is necessary to detennine the trajectories of particles. The two dimensionless, simultaneous equa­tions of motion for particles follow.

and

.1 (V:D- ~;;) 2 + (VrD- ~~;) 211/2

. (vrD-~~;)' ................... (12)

where

_(dVjP)I( dZ D)2 ( drD)21 Y2 N Rc - -- vD--- + vrD--- .

~ ~D ~D

.................................... (13)

These equations are derived from Newton's second law, and they state that the motion of a particle is governed by gravity and drag forces. The drag force acts on the parti­cle because of the local relative velocity between the par­ticle and the fluid.

The particle drag coefficient, K D, is a function of par­ticle shape and local Reynolds number and can be ob­tained from appropriate data on K D vs. NRc. 4 These published particle drag coefficient correlations are for fluids of constant viscosity, whereas the fluids employed in our experiments exhibit shear-rate-dependent viscosi­ty. However, since the drag force becomes significant only in the range of high fluid/particle relative velocity and since our fluids exhibit constant apparent viscosity above a shear rate of 300 seconds - I , we have elected to use the measured viscosity at 500 seconds - I in these correlations. All the velocities appearing in Eqs. II through 13 are ratios with respect to the free stream velocity, v I' which is the defined standard velocity. All calculated distances are ratios with respect to the casing radius, ri"' which is the unit of distance. The dimen­sionless time, t D, is defined in tenns of the casing radius, r c' the standard velocity, vI"' and the actual time, t, as tD=tXVtlrc. Thus, the unit of dimensionless time is the time required for a particle to travel a distance rc at a velocity VI.

862

Summary of Major Assumptions

The following simplifying assumptions were used to solve the particle trajectory problem.

I. The actual fluid motion was approximated by that of a frictionless, ideal fluid in a cylindrical segment with perforations along the vertex. Thus, the flow was steady, incompressible, irrotational, and axisymmetric.

2. The particles were considered as point masses that did not affect the flow field or interact with one another.

3. At large distances upstream of the perforations, the particles were distributed unifonnly and moved vertical­ly at their settling velocity with respect to the fluid.

4. The particles were monodisperse and spherical, and they experienced a drag force as in a real fluid.

Method for Solving the Equations of Motion

Eqs. II and 12 are second-order, nonlinear differential equations defining, at any instant, the accelerations of a particle in orthogonal directions. To find the particle­transport efficiency, we detennined the trajectory of a particle that just enters the bottom perforation. We then detennined the efficiency from the starting radial posi­tion of the critical trajectory (r *) by using a simple mass balance .

r 2

E=~ . ............................... (14) rM

We solved Eqs. II and 12 using a fourth-order Runge­Kutta method. We started the trajectory calculations at a distance of six casing radii above the top perforation. The initial axial velocity of the particle was set equal to the settling velocity of the particle at infinite dilution plus the free stream velocity. Increasing the distance upstream of the perforation did not affect the results within the third significant figure. The step length used in the calculations was checked similarly.

Comparison between Theoretical Results and Experimental Data Comparison of the theoretical transport efficiencies based on Eq. 14 and calculated trajectories, as just de­scribed, and experimental data are shown in Tables 1 through 3.

Table 1 compares theoretically derived transport effi­ciencies and the data of Torrest and Savage I for 20/40 U.S. mesh glass beads transported in water in a 5-in. casing with one perforation. Table 2 is a similar com­parison with some of our own experimental data obtain­ed in a 3-in. casing with one perforation. The 20/40 U. S. mesh gravel was transported with a hydroxyethyl cellulose solution having a viscosity of 66 cpo Table 3 compares predictions with data taken in a 3-in. casing with II perforations. In these experiments, 20/40 and 10120 U.S. mesh gravel was transported with tap water, and an equilibrium gravel bed extended from the bottom of the rat hole up to the bottom perforation.

Differences between theoretical and experimental results can arise from two sources: (1) the approximate nature of the theory and (2) the inherent inaccuracies of experimental measurements. In particular, we computed

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Exhibit 1013

TABLE 1-COMPARISON BETWEEN THEORETICAL RESULTS AND EXPERIMENTAL WORK OF TORREST AND SAVAGE1

Total Perforation Transport Efficiency'

Flow Rate Flow Rate (%)

(cu ft/sec x 10 2 ) (cu ft/sec x 10 2 ) Experiment Theory Difference

6.67 0.22 2.0 1.4 -0.6 6.67 0.44 3.1 3.0 -0.1 6.67 0.88 6.5 6.5 0.0 4.18 0.22 2.2 2.0 -0.2 4.18 0.44 4.5 4.4 -0.1 4.18 0.88 8.0 9.2 + 1.2 3.16 0.22 3.0 2.3 -0.7 3.16 0.44 5.4 5.1 -0.3 3.16 0.88 10.8 10.7 -0.1 2.60 0.22 3.0 2.7 -0.3 2.60 0.44 5.5 5.6 +0.1 2.60 0.88 11.0 11.6 +0.6 1.78 0.22 3.1 3.2 +0.1 1.78 0.44 6.6 6.7 +0.1 1.78 0.88 13.0 14.0 + 1.0 0.88 0.88 15.5 16.0 +0.5 0.44 0.44 8.4 8.7 +0.3 0.22 0.22 4.5 4.5 0.0

'Transport efficiency for 20140 U.S. mesh glass beads in water (5·in. casing, one perforation).

TABLE 2-COMPARISON BETWEEN OUR THEORY AND EXPERIMENTAL DATA-SINGLE PERFORATION

Total Perforation Transport Efficiency'

Flow Rate Flow Rate (%)

(cu ft/sec x 10 2 ) (cu It/sec x 10 2 ) Experiment Theory Difference 4,85 2.11 38.8 40.2 + 1.4 4.73 0.87 15.4 16.4 + 1,0 5,09 0,64 9,0 11.3 +2.3 4.40 0.27 4.7 4,9 +0,2 4.58 3,92 80,8 82,0 + 1.2

'Transport efficiency for 20140 U.S. mesh gravel in 66·cp HEC solution (3·in. casing, one perforation).

TABLE 3- COMPARISON BETWEEN OUR THEORY AND EXPERIMENT AL DA T A-11 PERFORATIONS

Total Perforation' • Transport Efficiency'

Flow Rate Flow Rate (%)

Gravel (cu ft/sec x 10 2 ) (cu ft/sec x 10 2 ) Experiment Theory

20/40 U.S. mesh 7.12 5,78 (top 10) 66 65 1,34 (bottom) 34 35

7,12 6.23 (top 10) 71 71 0.89 (bottom) 29 29

8.45 7,34 (top 10) 67 72 1.11 (bottom) 33 28

9.34 8.23 (top 10) 71 73 1.11 (bottom) 29 27

10/20 U,S, mesh 6.45 5.78 (top 10) 60 56 0.67 (bottom) 40 44

7.34 5,56 (top 10) 55 50 1.78 (bottom) 40 45

7.79 5.78 (top 10) 50 49 2.01 (bottom) 50 51

9.34 8.46 (top 10) 67 64 0.88 (bottom) 33 36

10.90 9.79 (top 10) 64 64 1,11 (bottom) 36 36

Difference

-1.0 + 1,0

0.0 0.0

+5.0 -5.0 +2,0 -2.0 -4.0 +4,0 -5,0 +5.0 -1,0 + 1,0 -3.0 +3,0

0.0 0.0

* Transport efficiency for gravel transported in tap waier (3-in. casing, perforations 2 in, apart, equilibrium bed at bottom perforation). * * Parentheses indicate flow rate in top 10 perforations or bottom perforation.

DECEMBER 1982 863 Page 7 of 9 Halliburton Energy Services, Inc.

Exhibit 1013

> '-'

.9

.8

z .7 w

~ u. u. .6 w f-a: ~ .5 V)

z ~

~ .4 w'

.3

. 2

.1

CASING DIAMETER ~ 6.2 IN .

PERFORATION DENSITY ~ 6 SHOTS/FT

NUMBER OF PERFORATIONS ~ 100

PERFORATION FLOW RATE ~qc;/100

O~~L-~L-~ __ ~ __ -L __ -L __ -L __ ~ __ ~ __ L-~ o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

qc;. FLOW RATE (FT3/SEC)

Fig. 10-Effect of flow rate on transport efficiency (calculated).

Fig. 11-Low-density gravel packs with void spaces are a result of radial inside-out gravel packing caused by use of high-viscosity fluids.

864

the theoretical efficiencies for an average particle diameter, whereas in the experiments a distribution of diameters existed. Despite these approximations, the computed results are in excellent agreement with measurements. In Table I the discrepancies between theory and the work of Torrest are random (eight theoretical values are slightly smaller than the measured values, eight are slightly larger, and two are identical). For transport of gravel in a fluid with a viscosity of 66 cp, the predicted efficiencies are all slightly larger than measured values; with this fluid, flow in the casing was laminar, so the assumption of an ideal fluid in the calculation of velocity fields could have been a major cause for the differences. The ideal fluid treatment neglects the effects of viscosity on the velocity field, and the flow pattern calculated differs markedly from a laminar flow pattern. Table 3 shows small random errors of the particle-transport efficiencies. Flow in this set of experiments was turbulent, but the small diameter of the casing and the fact that all the fluid exited the perfora­tions could have caused a flow pattern in the casing that was different from the axially symmetric, ideal flow pattern.

Simulation of Full-Scale Systems

The good agreement of theoretical estimates with ex­perimental data as just described suggests that particle­transport efficiencies can be predicted reliably for full­scale systems. Thus, the particle-transport efficiency of any full-scale system can be predicted by numerical solu­tion of the differential equations. Fig. 10 shows some theoretical predictions obtained for several fluid com­binations in a 6.2-in.-ID casing with 100 perforations. These results predict that significant increases in particle-transport efficiency can be achieved by increas­ing either the fluid viscosity or the total flow rate.

Summary and Conclusions The experimental and theoretical work described in this paper have given us some insight into particle transport and have let us derive some useful conclusions.

When particles are transported from the casing through perforations during gravel prepacking and hydraulic fracturing operations, inertial and gravity forces cause particle slurries to increase in concentration downstream of the perforations. This increase in particle concentra­tion can result in incomplete packing of voids outside the casing for either of two reasons: (I) particles can bridge in the entrance of perforation tunnels, and (2) highly concentrated slurries may not be transported through the perforations. Instead, the particles form a bed in the cas­ing until an equilibrium velocity is reached. At this velocity, the concentrated particle slurry is transported into the perforations.

To prevent bridging of particles in the perforations, the ratio of perforation diameter to average particle diameter should be 6: I or larger. To prevent a particle bank from forming in the casing opposite open perforations, design conditions that result in high particle-transport efficiency should be selected. The two design variables that can be varied most conveniently in prepacking operations are the pumping rate and the fluid viscosity. Increasing the viscosity of the carrier fluid to 40 cp can result in nearly

SOCIETY OF PETROLEUM ENGINEERS JOURNAL Page 8 of 9 Halliburton Energy Services, Inc.

Exhibit 1013

100 % particle-transport efficiency at moderate pumping rates of 0.2 to 0.3 cu ftlsec for most gravel prepack completions.

Note, however, that high-viscosity fluids should not be used for placing gravel between the screen and the casing when a circulating technique is used. In wellbores deviated less than about 45 ° when a tailpipe is lowered all the way to the bottom of the screen and low-viscosity (l to 6 cp) fluids are used to carry the gravel, the settling velocity of the gravel is sufficiently high that the par­ticles fall to the bottom of the well when the fluid enters the screen. The pack thus forms from the bottom of the wellbore upward. This type of bottom-to-top pack has a high density (about 90% of maximum achieved after mechanical packing) and offers the most effective sand control. By contrast, when fluids with high viscosity (e.g., greater than 40 cp) are used to carry the gravel or when the tailpipe is not lowered all the way to the bottom of the wellbore, the particles follow the fluid trajectories; as the fluid enters the screen, the pack forms from the outside of the screen radially outward to the casing. This type of inside-out radial packing generates large pressure gradients across the pack and screen, and eventually fluid circulation must stop. At this stage, the pack does not fill the annulus completely. When circulation stops, the gravel that was held on the outside of the screen by the fluid drag forces falls to the bottom of the well, leav­ing the upper portion of the screen uncovered and leav­ing voids in the part of the pack that covers the lower portion of the screen. Fig. 11 shows a photograph of a typical void left after the gravel falls following a gravel pack using a viscid carrier fluid. These voids certainly can allow invasion of formation sand into the gravel pack and can lead to early failure of the completion.

Nomenclature C = particle concentration in perforation,

Ibm/gal (kg/m3) C i = initial particle concentration in casing,

Ibm/gal (kg/m3) d = particle diameter, ft (m) E = particle transport efficiency, dimensionless g = gravitational acceleration, ft/sec 2 (m/s 2 )

KD = particle drag coefficient, dimensionless n t = total number of perforations, dimensionless

N D = drag number N Fr = Froude number NRe = Reynolds number, dimensionless q ci = initial volumetric flow rate in casing,

cu ft/sec (dm 3/s) q i = volumetric flow through perforation,

cu ft/sec (dm 3/s) r,Z = coordinates in cylindrical system, ft (m)

DECEMBER 1982

rc = casing radius, ft (m) rD,ZD = dimensionless coordinates in cylindrical

system r M = segment radius, ft (m) r * = radius of critical trajectory, ft (m)

t = time, seconds t D = dimensionless time v = fluid velocity vector, ftlsec (m/s)

Vj = free stream velocity, ftlsec (m/s) v r = radial velocity of fluid, ftlsec (m/s)

v rD = dimensionless radial velocity of fluid v, = settling velocity of particle at infinite

dilution, ft/sec (m/s) V z = axial velocity of fluid, ftlsec (m/s)

v zD = dimensionless axial velocity of fluid

'\j = vector differential operator, ft -I (m -I) () = azimuthal angle, radians

() M = angle of cylindrical segment, radians Jl = fluid viscosity, lbm/ft-sec (Pa' s) p = fluid density, lbm/cu ft (kg/m3)

p" = particle density, lbm/cu ft (kg/m3) <I> = velocity potential, sq ftl sec (m 2/ sec)

Acknowledgments We express our appreciation to the management of Exxon Production Research Co. for permission to publish this paper.

References l. Torrest, R.S. and Savage, R.W.: "Particle Collection from Ver­

tical Suspension Flows Through Small Side Ports-A Correla­tion," Cdn. f. ChE. (Dec. 1975) 53, 699.

2. Haynes, C.D. and Gray, K.E.: "Sand Particle Transport in Per­forated Casing," f. Pet. Tech. (Jan. 1974) 80-84.

3. Morse. P.M. and Feshbach, H.: Methods o/Theoretical Physics. Part 2, McGraw-Hili Book Co. Inc., New York City (1953).

4. Bird, R.B., Stewart, W.E., and Lightfoot, E.N.: Transport Phenomena, John Wiley and Sons Inc., New York City (1965) 192.

SI Metric Conversion Factors cp x 1.0* E-03 Pa's

cu ft x 2.831 685 E-02 m3

ft x 3.048* E-Ol m OF (OF-32)/1.8 °C

gal x 3.785412 E-03 m3

in. X 2.54* E+OO cm Ibm X 4.535924 E-Ol kg

*Conversion factor is exact. SPEJ

Original manuscript received in Society of Petroleum Engineers office Dec. 19, 1977. Revised manuscript received Jan. 30. 1979. Paper accepted for publication Aug. 26, 1980. Paper (SPE 7006) first presented at the 1978 SPE Formation Damage Control Symposium held in Lafayette, LA, Feb. 15-16.

865 Page 9 of 9 Halliburton Energy Services, Inc.

Exhibit 1013