SPE 96339

An Integrated Transport Model for Ball-Sealer Diversion in Vertical and Horizontal Wells X. Li, SPE, Z. Chen, SPE, S. Chaudhary, SPE, and R.J. Novotony, SPE, BJ Services Co.

Copyright 2005, Society of Petroleum Engineers This paper was prepared for presentation at the 2005 SPE Annual Technical Conference and Exhibition held in Dallas, Texas, U.S.A., 9 12 October 2005. This paper was selected for presentation by an SPE Program Committee following review of information contained in a proposal submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to a proposal of not more than 300 words; illustrations may not be copied. The proposal must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435. Abstract Ball sealer diversion has been proven to be both an effective and economic way to selectively stimulate low permeability oil and gas reservoirs in hydraulic fracturing and matrix acidizing treatments. However, the design and implementation of a successful ball sealer diversion treatment is still a challenge. Often the designer depends on experience, and lacks the knowledge of accurate ball transport and sealing behaviors. An integrated model for selecting operating factors such as fluid and ball properties, as well as predicting the ball sealers transport and hydraulic behavior prior to pumping is needed for optimizing the stimulation process. In this paper, an integrated transport model is presented to describe the relationships among the ball sealer transport sealing behavior, wellbore deviation, wall effect, perforation density and size, fluid properties, pumping rate and ball properties. In addition, the smoothness of ball, perforation phasing, and velocity profile inside the wellbore during ball seating are also taken into consideration. Recommendations are provided for determining the number of ball sealers per job for either single or multiple stage treatment, the designed pumping rate, and the physical properties of the fluid and ball sealer. A hydraulic analysis model is presented for the overall fluid dynamics starting from surface, through wellbore, to reservoir. This analysis describes the effects of reservoir condition, pressure drop on perforations, and actual sealing efficiency on the surface treatment pressure profile. This paper will investigate the effects of the diversion factors on the ball transport behaviors such as transport time, ball sealer efficiency and surface pressure. Introduction One of the first ball sealer process was performed by the Western Company in 1956[1]. Since then, it has been widely

applied in the selective well treatment and stimulation[2]. During this ball sealer process development, a significant improvement was reported by using near-neutral buoyancy balls instead of conventional ball sealers. It was reported that the buoyant ball sealer seating efficiency was increased to 100%[3,4]. Although there are several references that have studied the properties of this diversion process, the implementation of this technique in the industry is still relies on the field experience and rules of thumb. Modern techniques such as Laser Doppler Anemometry (LDA) and Particle Image Velocimetry (PIV) provide useful measurement tools to observe transport behaviors in the solid particle-fluid system. And advanced CFD simulators offer numerical analysis in the fluid dynamics. However, the mechanism of transport phenomena in the particle-fluid system is still not fully understood. Hydrodynamic forces and moments acting on particles enclosed by fluids depends on many factors, such as the local (undisturbed) flow field, fluid and particle inertia, external forces and particle shape. Non-Newtonian fluids and turbulent flow increase the degree of the problems complexity. For this diverting process, these factors are fluid and ball properties, pumping rate, well geometry and perforation data. In order to understand this transport process, an integrated transport model was developed to describe the relationships among these factors and to develop a software product as a design and simulation tool for guiding the implementation of the process. Transport Model Based on the principle of fluid mechanics, the ball tracks are simulated through a Lagrangian approach taking into account only the one-way coupling effect. The transient model for this transport process is an ordinary differential equation with an initial condition. The equation of ball motion can be expressed as:

dSnTgdtud

SBB

BBB VV

+= (1) where VB is the volume of the ball , B is the density of the ball, uB is the ball velocity vector, and T is the instantaneous stress tensor that must satisfy the Navier-Stocks equations.

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In Equation (1), the stress tensor term can be described as[5]:

+++= LAPGDS

FFFFdSnT (2)

where subscripts D, PG, A and L respectively denote force components arising from drag, gravity, flow pressure gradient, added mass effect, and lift force. When the ball achieves a steady-state condition, the balls velocity becomes a constant, this velocity is called terminal settling velocity (or slip velocity). The ball will be acted on by two main forces when it closes to the perforation. These two main forces are an inertial force whose direction is along the axis of casing and a drag force toward to the perforation. The ball seating depends mainly on the magnitude of these two forces. If the drag force is greater than the inertial force, the ball will hit the perforation. Otherwise, it will move down to the next perforation and the evaluation continues. If the ball has not seated before passing all the perforations in the interval, the ball will drop to the bottom of the well. The inertial force is a function of terminal settling velocity and wall effect. And the drag force toward perforation is a function of velocity profile from the center of the casing to the entrance of perforation and discharge coefficient. The other criterion is related to the ball holding force after the ball seats on the perforation. This holding force is created by the pressure difference across the perforation. The force trying to dislodge the ball is a drag force caused by the main stream flow of fluid. For example, In the case of vertical well, if the ball unfolding force is greater than the vertical component of the ball holding force, the ball will drop. Therefore, the successful sealing of the perforation depends on two main factors: seating and holding. Only a ball that can meet both two criteria is considered seated. In order to predict the surface pressure as a function of time during this diversion process, the perforated zones reservoir information is taken into consideration of this integrated model. This information provides more details for hydraulics analysis. Numerical Methods The ordinary differential equation following a Lagrangian particle tracking methodology was solved numerically by a fourth-order Rung-Kutta integrating method. Hence, the instantaneous positions and velocities of the spherical ball can be calculated numerically. The balls terminal velocity is a function of drag coefficient. And the drag coefficient is a function of Reynolds number that is related to the terminal velocity. In order to obtain the velocity, a trial-and-error solution is adapted to deal with this coupling problem.

The drag coefficient is calculated from the following equations[6]: For Newtonian fluid: Cd= 24/Re Re

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where Pwf(t) : flowing pressure at center of top zone; Pi : reservoir initial pressure; Q : pump rate, BPM; : fluid viscosity, cp; B : fluid volume factor; n : the number of unsealed perforations; N : total number of perforations of this zone; k : reservoir permeability, Md; h : reservoir thickness, ft; re(t) : radius of drainage at time t, ft rw : wellbore radius;

( ) [ 5.0/0037.0 te ckttr = ] (6) where : porosity, %; t : pump time, min; ct : total compressibility, 1/psi; The surface treating pressure is obtained by adjusting bottomhole pressure with frictional and hydrostatic pressures. The surface increase delta P after pumped a time t is: ( )NknhrtrQuBppP wesurfsurftsurf //]/)((ln[2033280 ==

(7)

For spherical particles, there are well over 30 equations in the literature relating to the drag coefficient (Cd) to the Reynolds number of spheres falling at their terminal velocity. Some correlations are of varying complexity, and contain as many as 18 arbitrary parameters. In this paper, we use the model as described above as we consider the balls sphericity is 1.00. However, For the higher Re regimes, we obtained polynomial functions by regression analysis with the Achenbachs data, and use these functions for the calculation.

Surface pressure changes during the ball sealer process Figure 7 describes a typical tendency about pressure change as the perforations are sealed. From this figure, at a certain time, balls start to seal the perforations, the pressure increases slightly compared to the pressure when all perforations are opened. Once the open perforations percentage decreases to ~20%, the pressure increases dramatically. Simulator The ball sealer simulator is developed on Microsoft Windows platform with friendly graphical user interface. It can cope with both conventional and buoyant ball sealers. The simulator is also adjusted to fit the practical application on the basis of experimental data. Several main features of this simulator are listed as below: Newtonian and Non-Newtonian fluids; Single and multiple stages treatment; Wall effect on the ball transport behavior; The effect of well geometry (Vertical, inclined and

horizontal); The influence of fluid and ball properties; Effect of perforation data (Perforated zone length,

perforation numbers, shot density, perforation size and phasing etc.);

The effect of pump rate. The simulation results are presented with several useful information such as predicting the balls seating and holding status, ball sealer and fluids transport time to the specified perforation, fluid volume used for the ball sealer treatment and how many perforation are sealed after the treatment.

In addition, the simulator conducts hydraulics calculation and provides the results of surface pressure as a function of time during the ball sealer process. Results and Discussion Unsteady period and Settling Velocity The balls velocity as a function of time is shown in Figure 1.In this figure, we find that the time for the ball to achieve terminal setting or rising velocity is short. For example, in these cases, it is just a few seconds. This indicates that it is reasonable that the actual falling velocity is considered a constant in the wellbore during the ball sealer process. The simulators accuracy and computing efficiency depend mainly on the time step length (t) during the simulation. Mathematically, we have to choose t -> 0 so that dV/dt become more accurate. However, this will increase the expense of computational time. In reality, some balls may not seat on the perforations during the process. These balls will be present in the fluid. The velocity gradient around each ball may be effected by the presence of nearby balls. This is a hindered settling. In this paper, we will not consider this factor in our model. Drag coefficient and ball surface smoothness

The effect of balls surface smoothness on the ball settling velocity is shown in Figure 2. The difference between smooth ball and dimpled ball is mainly from drag coefficient. It should be noted that the effect of smoothness is in a certain Res range, typically from 20,000 to 300,000. From Figure 2, it shows that the terminal settling velocity of a dimpled ball is larger than that of a smooth ball. From fluid mechanics viewpoints, the drag force experienced by a ball as it moves through a fluid is usually divided into two components: viscous drag and pressure drag. Viscous drag is associated with the viscous stresses developed within the boundary layers while pressure drag comes from the eddying motions that are set up in the wake downstream of the ball. At a very low Reynolds number, the drag exerted on the ball is predominately viscous drag. The pressure distribution on the ball in this region does not contribute much to the overall drag. While at large Reynolds number, pressure drag is the dominant form of drag present. Viscous effects are still present since the fluid has a viscosity, but its effects remain in the thin boundary layer next to the ball. According to the boundary layer theory[9], the fluid at a moderate rate, reaches a separation point which is a position on the ball surface that the velocity gradient becomes zero. In other words, the adverse pressure gradient causes the fluid to separate from the surface at this point on the ball. As the velocity increases, the

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flow of fluid is therefore reversed in this area producing eddies and formation of a wake region behind the ball. In the case of a smooth ball, the boundary layer on the surface is experienced laminar flow, this boundary layer tens to separate and break away from the ball easily. This separation of the boundary layer causes a large wake of low pressure behind the ball, resulting in a high drag force. On the other hand, in the case of a dimpled ball, the dimpled surface makes the boundary layer more turbulent, this causes the boundary layer to extend further along the ball, resulting in the wake behind the ball is smaller than that behind a smooth ball. The smaller the wake, the lower pressure and the easier it is for the fluid to flow around the ball and so the drag force is less. Perforation Density and Phasing For many stimulation treatments, the perforation density is usually in the range of 1~12 shot/feet. We examine a test case (Test Case A) with a perforated zone of 20 feet. The pumping fluid is 15%HCl with 5 bpm pump rate for a single-stage ball sealer process. The RCN (Rubber Coated Neoprene) balls size is 7/8 with specific gravity is 1.3. The perforation size is 0.375 and phasing is zero degree. The casing I.D. is 4.0. The ball sealer efficiency as a function of shot density is shown in Figure 3. The ball sealer efficiency is defined as the number of sealed perforations divided by total number of perforations. In this example, for the shot density below 3 shots/ft, the ball sealer efficiency is 100%. It means that under the process conditions (pump rate, fluid and ball properties, perforation size etc.), the number of perforations in this perforated interval is suitable for meeting the criteria of both seating and holding. As the shot density increases, less fluid goes through each perforation to the reservoir, the fluid velocity in the mainstream is reduced, this causes the decrease of the inertial energy of ball in the fluid. Eventually it decreases the ball sealers efficiency. In this case, when the shot density is over 8 shots/ft, the ball sealer efficiency becomes zero, which means the ball sealer is total failed. This useful information will provide a guide for the operation engineers in design the ball sealer process. Figure 4 shows the relationships among the non-seated ball percentages, and shot density. In this example, the dropped ball numbers are the same as the total perforations. We define the non-seated ball percentage is the number of non-seated balls divided by the total number of dropped balls. In this example, the non-seated percentage is increased as the shot density is increased. Figure 5 shows the relationship between the phasing and ball transport time. We use the above test case with the shot density is 2 shots/ft. The ball transport time is the time necessary for a ball to seal at the last perforation. There are very few referemces in the literature on studies of the phasing effect on the particles transport trajectory in the fluid inside the wellbore. Although Chen et al[10] proposed some mathematical treatments for flow efficiency of perforated systems in the wellbore. Their work was more theoretical and the systems were not for solid particle multiphase fluids.

From this figure, it turns out the phasing has little effecs on the ball transport time in the casing. In general, the ball trajectories in the fluid are fairly complicated because both translational and rotational motions should be simultaneously considered and these motions are coupled. To simplify the simulation process, we assume that the ball trajectory follows a certain pattern along the fluid transport. Statistically, the instantaneous balls transport velocity consists of the mean velocity and a fluctuating velocity. Under the same Reynolds number and other conditions, the perforations phasing orientation may not change the local disturbance or perturbation. On the other hand, the perforation phasing does not change the balls transport length within the shot density of 1~12 shots/ft. Hence, the phasing has a little effect on the balls transport behaviors. Ball transport time Table 1 lists partially data that present the relationship between transport time and wellbore measured depth. Internal experiments show that the ball transport time in reality is longer that that of this simulator. The reasons for this behavior may be explained as follows. From Equation (2), we could find that the forces on the ball when the ball is moving through a fluid are quite complex. As we discussed in the above section, for a fast moving ball, there will be some point on the ball where the flow separates, creating a turbulent wake behind the ball. As the pressure differential between the leading surface of the ball and the surface within the turbulent wake arrives at a critical point, it may cause the ball to migrate laterally and spin. So the spinning of the ball in the fluid could result in the retarded ball in the transport time. The spinning of solid particles in fluid is a very complex topic in physics of fluids, which is out of the scope of this paper. The ball spinning will cause a lift force for the ball. This is so called Magnus effect. This lift force is usually perpendicular to the direction of the fluid flow and related to the angular velocity of the fluid flow. This force is not easy to calculate analytically in the fluid dynamics. Although Joseph et al[12] proposed a DNS (direct Numerical simulation) method to compute the slip velocity in the low Reynolds number range with the lift force is taken into account, there still no experimental data to confirm it. Overall, both the lateral migration and lift force may cause the retarded movement of ball along the centerline in the casing and then increase the balls transport time. Inclined and Horizontal Well Figure 6 shows the effect of wellbore inclination on the ball sealers performance. In this test case (Test Case B except the shot density is 4 shots/ft), the ball sealer efficiency increases as the wellbore inclination angle increases, while the non-seated ball percentage decreases as the inclination angle increases. Under the same conditions, the inclination angle mainly changes the settling velocity of ball, which changes ball inertia force, ball diverting force, as well as the holding force.

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Conclusions 1. An integrated transport model is developed on the basis of Lagrangian approach with one-way coupling effect in the fluid dynamics. 2. This ball sealer simulator is capable of dealing with both Newtonian and Non-Newtonian fluids. Several main factors such as wall effect, wellbore geometry, fluid and ball properties, perforation shot density and pump rate are taken into account in this software. 3. The simulation results show that the time period from unsteady state to the steady state is quiet short. 4. The test cases in this paper demonstrate that the ball sealer efficiency decreases as the perforation shot density increases, while the ball sealer efficiency decreases as the perforation discharge coefficient increases. 5. Drag coefficient, surface smoothness, and wellbore inclination angle play an important role in the ball sealer diverting process. 6. This simulator also provides hydraulic anlysis integrating with reservoir information during the ball sealer process. Nomenclature API = American Petroleum Institute B = fluid volume factor CFD = Computational Fluid Dynamics Cp = discharge coefficient ct = total compressibility, 1/psi DNS = Direct Numerical Simulation h = reservoir thickness, ft k = reservoir permeability, Md LDA = Laser Doppler Anemometry N = total number of perforations of this zone n = the number of unsealed perforations Pi = reservoir initial pressure PIV = Particle Image Velocimetry Pwf(t) = flowing pressure at center of top zone Q = pump rate, BPM RCN = Rubber Coated Neoprene re(t) = radius of drainage at time t, ft rw = wellbore radius T = the instantaneous stress tensor t = pump time, min VB = the volume of the ball

B = the density of the ball = fluid viscosity, cp = porosity, % References 1. Harrison, N.W.: Diverting Agents Their History and

Application, J. Pet. Tech. (May 1972)593-598. 2. Brown, R.W., Neill, G.H., Loper, R.G.: Factors

Influencing Optimum Ball Sealer Performance, J. Pet. Tech. (April 1963) 450-454.

3. Bale G.E.,: Matrix Acidizing in Saudi Arabia by Using Buoyant Ball Sealers, J. Pet. Tech. (October 1984)1748-1753.

4. Erbstosser, S. R.: Improved Ball Sealer Diversion,J. Pet. Tech. (Nov. 1980)1903-1910.

5. Babiano, A., Cartwright, J. H. E., Piro, O., Provenzale, A.: Dynamics of Small Neutrally Buoyant Sphere in a Fluid and Targeting in Hamiltonian Systems, Phys. Rev. Lett.,(June 2000)5764-5767.

6. Clift, R., Grace, J. R., Webber, M. E.: Bubbles, Drop and Particles, Academic Press, New York, 1978.

7. Felice, R., Gibilaro, L. G., Foscolo, P. U.: On the hindered Falling Velocity of Spheres in the inertial Flow Regime, Chem . Engr. Sci., (1995)30053006.

8. Govier, G. W., Aziz, K. The Flow of Complex Mixturesin Pipe, Litton Educational Publishing Inc., New York, (1972)4-13.

9. Bird, R.B., Stewart, W. E., Lightfoot, E. N.:Transport Phenomena, John Wiley and Sons Inc., 2002.

10. Chen, C. Y., Arkinson C.:Flow Efficiency Of Perforated Systems- A Combined Analytical And Numerical Treatmnent,(2001), J. of Engr. Math., 159-178..

11. Patankar, N.A., Huang, P.Y., Ko,T., Joseph D.D. : Lift-off of a single particle in Newtonian and viscoelastic fluids by direct numerical simulation,(2001), J. Fluid Mech., 438, 67-100.

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Ball Velocity versus time

01020304050607080

0 0.5 1 1.5 2 2.5Time, s

Velo

city

, ft/m

in 7/8" Ball,S.G.=1.3,Settling Velocity7/8" Ball,S.G.=1.1,Settling Velocity"7/8" Ball,S.G.=0.9,Rising Velocity

Figure 1. Ball Velocity as a function of time

Comparison between dimpled ball and smooth ball (7/8", S.G.=1.3, Settling velocity)

0

20

40

60

80

100

0 0.5 1 1.5 2 2.5

Time, second

Velo

city

, ft/m

in

Dimpled ballSmooth ball

Figure 2. Ball smoothness effect on the settling velocity

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Ball Sealer Efficiency vs. Shot Density

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12

Shot Density, #Shot/ft

Effic

ienc

y(%

)

Figure 3. Efeect of Shot Density on Ball Sealers efficiency

Precentage of unseated or unholded perforations vs. Shot density

00.10.20.30.40.50.60.70.8

0 1 2 3 4 5 6 7 8 9 10 11 12

Shot density, #shots/ft

Uns

aete

d or

unh

olde

d pe

rfor

atio

ns, % Unseated

Unholded

Figure 4. Effect of Shot Density on unseated ball or unholding ball percentage

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Transport time vs. Phasing

458

460

462

464

466

468

470

0 50 100 150 200

Phasing

Tran

spor

t tim

e, s

econ

ds

Figure 5. Effect of Phasing on balls transport time

Wellbore inclination vs. ball sealer efficiency and unholded ball percentage

010203040506070

0 20 40 60 80 100

Wellbore inclination Angle

%

Ball Sealer Efiiciency

Unholded ball Percentage

Figure 6. Relationship between wellbore inclination and ball sealer efficiency

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Surface Pressure Response

0

100

200

300

400

500

600

700

800

900

1000

0 5 10 15 20 25 30 35

Pump Time

Pres

sure

Incr

ease

0

10

20

30

40

50

60

70

80

90

100

Ope

n pe

rfor

atio

ns, %

All Perfs OpenSome Perfs SealedOpen Perfs

Figure 7. Surface pressure as a function of pumping rate

Table 1. Ball transport time (partially list)

Perforation Depth (M.D., ft)

Ball Sealed Status

Ball Transport Time (Seconds)

5830.50 No(Unhold) 5830.75 No(Unhold) 5831.00 No(Unhold) 5831.25 No(Unhold) 5831.50 Yes 1449.7 5831.75 Yes 1449.8 5832.00 Yes 1450.0 5832.25 Yes 1450.2 5832.50 Yes 1450.4 5832.75 Yes 1450.6 5833.00 Yes 1450.8 5833.25 Yes 1451.0 5833.50 Yes 1451.1