euler-lagrange numerical model for the sulfate scale ...enahpe2019.ipt.br/arquivos anais do...

Enahpe 2019 - 015 ENAHPE 2019 – Encontro Nacional de Construção de Poços de Petróleo e Gás

Serra Negra – SP, 19 a 22 de Agosto de 2019

Euler-Lagrange numerical model for the sulfate scale

formation including the effects of adhesion and solid

crystal growth

Marina E. Mazuroski1, Vinicius G. Poletto1, Fernando C. De Lai1

André L. Martins2, Silvio L. M. Junqueira1

1Research Center for Rheology and Non-Newtonian Fluids – CERNN, Federal University of Technology - Paraná –

UTFPR, Curitiba-PR, Brazil, 81280-340,

[email protected], [email protected], [email protected], [email protected] 2Research and Development Center – CENPES, PETROBRAS, Rio de Janeiro-RJ, Brazil, 21941-915,

[email protected]

Abstract Scale deposition in downhole equipment and completion systems can cause productivity issues, with limited oil output,

equipment damage and safety issues. In this context, the experiment Dynamic Tube Blocking test, for instance, allows

the studying of crystal formation under different saturation, temperature and pressure conditions. However, the DTB,

like other related experiments, are run with low flow rates, which are far from representing the real downhole flow

environment. Furthermore, limitations arise in determining the scaling tendency of complex geometry parts such as

sand control screens and valves. This work proposes a numerical model for simulating liquid-solid flow for the scaling

formation process, incorporating the effects of crystal growth and adhesion. The solution is represented as a

continuous fluid phase evaluated through an eulerian approach, while the crystals in solid phase are portrayed as

discrete spherical particles with a lagrangian approach. Both are mathematically formulated with the Dense Discrete

Phase Model (DDPM) coupled with the Discrete Element Method (DEM), resulting into a four-way interaction

between the phases. The growth mechanism is simulated through User Defined Functions so that different particle

functions for the diameter growth are studied in a capillary tube as well as on a heterogeneous porous medium. The

constant force model represents adhesive forces, assessed through a repose angle study and then incorporated in the

DTB simulation. Further calibration of the numerical model will allow the prediction of the flow reaction to scale

formation phenomena like crystal growth and solids adhesion followed by deposition on a surface.

1. Introduction Secondary oil recovery can be achieved through

injection of seawater into production wells, which

maintains reservoir pressure and increases the oil

output. However, the mixture of incompatible seawater

and formation connate water triggers scale formation

and deposition, one of the most serious problems that

can affect hydrocarbon production environments.

Scaling decreases the productivity of the well through

the formation of a thick layer that can take place in

injection and production wells, pipelines, other

production facilities and equipment, causing blockage

and clogging the flow, severely increasing the pressure

drop in the production tubing. Precipitation can also

cause formation damage in the reservoir and increase

corrosion rates, resulting in safety issues to the

operation [1], [2].

The deposition process initiates because of the low

solubility of substances formed through the mixture of

injection and formation water containing

complementary salt ions. Combined with certain

temperature and pressure conditions of the system, and

influenced by other parameters, these salts may achieve

a supersaturated state, and precipitation prospect

increases drastically [3].

Depending on the reservoir and seawater compositions,

different types of scale can assemble, the most common

being sulfate and carbonate salts, such as barium

sulfate, strontium sulfate, calcium sulfate and calcium

carbonate [3].

At certain circumstances, an effective scale

management strategy helps assuring oil production

conditions. Most measures focus on reducing scale

potential, since the removal of deposited salts from

down hole equipment has a much higher cost [1], [4].

Common measures to prevent scale formation include

desulfation of injected water, which avoids only sulfate

scale, and use of inhibitors, chemicals designed to

restrain crystal formation and growth [1], [2]. The

classification of the inhibitor relies on the mechanism

of the scale prevention: Thermodynamic inhibitors act

by decreasing the ionic activity and therefore



decreasing the super saturation of the solution. On the

other hand, kinetic inhibitors act on the adsorption of

the solid crystal surface, hindering the incorporation of

ions and, as a consequence, its growth [4].

The effectiveness of chemical inhibitors is influenced

by the thermodynamic conditions of the well,

composition of the water mixture, thermal stability of

the inhibitor and brine compatibility [5]. Mostly

important, the action of the inhibitors is only

accomplished by the injection in the seawater-brine

mixture at the downhole assembly. However, the

production zones in modern well completion systems

are not approachable by the injection lines, remaining

vulnerable. For instance, it is not possible to use

injection lines in Petrobras’s cableless intelligent well

completion design [6].

The downhole parts (e.g. internal control valves, sliding

sleeves valves, sand control screens, gravel packing)

are prone of inorganic scaling. For this reason, the

understanding of scale formation has been the subject

of studies for decades. Thermodynamic and kinetic

models are widely available in commercial softwares

such as MultiScale, GWB and SOLMINEQ. However,

while such computer packages are able to determine the

saturation of a system based on thermodynamic

conditions, they do not take into account the influence

of flow dynamics over the scale formation. Therefore,

miscalculations of the scaling risk arise

On the other hand, experimental tests are appropriate

for studying the hydrodynamic effects on scale

precipitation, and are usually combined with the results

from thermodynamic modelling. For instance, the

Dynamic Tube Blocking Test (DTB) is a common

experiment for evaluating scale formation dynamics in

a capillary tube by measuring the pressure drop

response of the flow of a supersaturated ionic solution.

Spite of the possibility of setting high pressure and high

temperature in the DTB, the flow rate is low, not

representing well production conditions.

Numerical modelling of the process of scale formation

may arise as a solution for the evaluation of the scaling

tendency in downhole parts with complex geometry,

like sand screens and SSV valves. Furthermore, the

numerical method is capable of implementing a more

extensive range of boundary conditions to represent

production conditions, such as high temperatures and

high pressures, which are difficult to manage in

experimental tests.

In this work it is described the developments into a

numerical model for the liquid-solid two-phase flow for

the simulation of the process of scale formation,

incorporating the effects of particle growth and particle

adhesion. A hybrid Euler-Lagrange approach is applied

for the formulation of the flow and the solid particles,

which mimic the precipitated sulfate crystals. The four-

way interaction between the phases allow the

evaluation of the particle effect over the flow field. The

numerical simulation is resorted by coupling the Dense

Discrete Phase Model (DDPM), able to compute either

dilute or dense particulate flows, to the Discrete

Element Method (DEM). The DEM incorporates the

influence of forces that arises from collision, friction

and adhesion among particles or between a particle and

a surface. In this work, a number of tests are presented

to explore the potentiality of the DDPM-DEM, like the

simulation of the particle growth in a capillary tube and

in a heterogeneous porous medium. The particle

adhesion is studied through the reproduction of the

angle of repose test and the adhesion into a capillary

tube.

2. Problem Formulation The scale formation process is envisioned as a two-

phase liquid-solid flow constituted of a liquid medium

with discrete solid particles, which mimic the

precipitated crystals of sulfate. The crystal formation

process is illustrated in Figure 1, which begins with a

supersaturated solution with the formation of small

nuclei (approximately 10-10 m). These nuclei will grow

or undergo secondary nucleation, which starts from

already existent nuclei. If the system conditions allow,

the clusters will grow to larger sizes (100 µm), and start

to agglomerate and adhere to other surfaces, constantly

breaking up and forming new groups. The numerical

model proposed in the present work comprises the

phenomena of crystal growth, agglomeration/adhesion

and crystal breakup.

Figure 1. Two-phase liquid-solid flow for the scaling

process including adhesion and growing effects.

The liquid-solid two-phase flow is formulated through

an Euler-Lagrange approach. The equations are

numerically simulated with the CFD-DEM coupling by

the Dense Discrete Phase Model (DDPM) [7] and the

Discrete Element Method (DEM) [8]. The mass and

momentum balance equations are adapted to

incorporate the influence of the solid phase through the

means of the fluid volume fraction (εβ), as seen in Eqs.

(1) and (2), and the momentum coupling term, Fpβ



[N/m³]. Such equations are able to model either dilute

or dense particulate flows.

0t

u (1)

T

p

t

p

u u u

u u g F

(2)

The solid phase is represented by discrete spherical

particles individually tracked in Lagrangian referential,

with the position xp individually calculated from the

velocity up in Eq. (3), which is obtained by the linear

momentum balance in Eq. (4). The particle angular

momentum balance is also calculated, through Eq. (5).

[ ]

[ ]

p j

p j

d

dt

xu (3)

[ ]

[ ] [ ] [ ] [ ]

[ ] [ ]

p j

p D j G j VM j PG j

SL j DEM j

dm

dt

uF F F F

F F

(4)

[ ]

[ ] [ ]

p j

D j DEM j

dI

dt

ωT T (5)

The forces in Eq. (4) arise from the flow effects, such

as the drag force, the virtual mass force, the pressure

gradient, the Saffman lift and the Magnus lift. The

particle-particle or particle-wall interactions result in

collision, frictional or adhesion forces. The torques,

from Eq. (5), represent the drag torque and the contact

torque, respectively. The equations for these forces are

summarized in Table 1.

Table 1. Forces acting on particles.

Forces

Drag [ ]

[ ] [ ] [ ] [ ]2

Re18( )

24

D p j

D j p j j p j

p p

Cm

d

F u u (6)

Gravity and buoyancy

[ ] [ ]

p

G j p j

p

m

gF (7)

Virtual mass [ ] [ ] [ ] [ ]VM j VM p j j p j

p

DC m

Dt

F u u (8)

Pressure gradient [ ] [ ] [ ] [ ]PG j p j j j

p

m

F u u (9)

Saffman lift [ ] [ ] [ ] [ ] [ ]( )SL j SL p j j j p j

p

C m

F u u u (10)

Contact

[ ] [ , ] [ , ] [ , ] [ , ]DEM j n i j i j t i j i j F F n F t

[ ] [ , ] [ , ]n j n n n p i j i jk F u n

[ ] [ ]t j a n jF F

,

0 if

min( , ) if

n adh

n adhadh n adhi j

s

f g m m s

F

(11)

(12)

(13)

(14)

Torques

Drag [ ] [ ] [ ]d j j p jC T ω

1/2 1

[ ] [ ] [ ]6,45Re 32,1Rej j jC

(15)

(16)

Contact [ ]

[ ] [ ] [ ] [ ]

[ ]2

p p j

DEM j ij t j r n j

p j

d

ωT n F F

ω (17)

The fluid and particulate phase numerical solution are

calculated separately, initiated with the convergence of

the fluid phase by finite volume method, followed by

the solid phase injections into the domain. The particles

velocity and angular velocity are individually

determined by implicit numeric integration, with the

particle-fluid forces calculations based on the fluid

flow field. Their position is then estimated through the

Crank-Nicholson scheme. Collision and adhesion

forces are computed only upon identification of contact

among particles between a particle and a surface, and

fulfillment of adhesion criteria.



The particles in a collision are treated are soft-spheres

that can overlap each other. The force generated can be

calculated through various models, with the spring-

dashpot system being a simple and effective one,

described by Eq. (12). It takes into account the particles

spring coefficient kn [N/m] damping coefficient ηn

[N.s/m] and the superposition extent [7]. The friction

foces are calculated through the Coulomb model,

defined in Eq. (13).

The adhesion effects are mainly calculated through the

contact force achieved during collision. Several

adhesion force models are proposed in literature, with

different input variables. The simplest one presented is

the constant adhesive force model, which activates

based on a determined minimum distance δadh [m]

between particles or particles and boundaries, inflicting

a constant force value Fn,adh [N] based on the solids

mass [8]. The model is described by the Eq. (14).

The momentum balance equation then receives the

solid and fluid volumetric fraction, and the volume-

averaged fluid-particle forces in each mesh control

volume, followed by a new solution of the continuous

phase by the finite volume method, new particle

injections, the update of the particle forces and so on.

3. Results and discussions The DDPM-DEM numerical model capabilities are

demonstrated though reproduction of literature results

The fluid-particle interaction forces are evaluated by

comparing the experimental results of Mordant and

Pinton [9] of beads abandoned from rest in water. The

numerical results for acceleration curves reaching

terminal velocity are presented and compared with

numerical results in Figure 2. The calculation of the

fluid-particle forces is accurate.

Figure 2. Comparison of simulated (DDPM) and

experimental [9] results for the acceleration curves of beads

abandoned in water.

The computation of the particle’s collisional force

imping normally on a surface is also verified, since

such force affects the adhesion mechanism included in

the numerical model. Gondret, Lance and Petit [10]

showed experimental results for particle rebound

acceleration, which is correlated to the numerical

calculations obtained via DDPM-DEM. The

comparison, as seen in Figure 3, indicates good

agreement and reinforces the numerical model capacity

of accurately reflecting wall effects into solid discrete

particles.

Figure 3. Comparison of DDPM-DEM numerical and

experimental [10] results for the acceleration curve of a

single particle impinging normally on a surface normal.

Finally, the scale formation process is considered as a

particle deposition into a surface followed by a particle

bed formation. Therefore, a dense solid two-phase flow

is observed, which necessarily affects the fluid flow by

imposing a pressure drop. To fulfill such verification,

experimental results for the transient bed formation of

a liquid-solid flow in a horizontal channel are displayed

in Figure 4. The numerical result reproduces the bed

formation process, resulting in a bed of comparable

dimensions to the experimental one. Furthermore, at 50

s, the fluid velocity field indicates that the fluid

accelerates above the bed, corroborating the particle’s

effect into the flow field.

Figure 4. Reproduction of literature results for the

horizontal liquid-solid flow with bed formation.

Dynamic Tube Blocking Test simulation parameters

were based on Santos [11] experimental set up with



water flowing in laminar regime through a 0,5 mm tube

with length of 1,0 m, which is shortened to 0,0625 m to

reduce the mesh size. The solid phase is considered

mono disperse with constant diameter and properties,

cast in accordance to that of barium sulfate crystals.

The main parameters are listed in Table 2.

Table 2. Numerical test parameters

Tube diameter dt (mm) 0,5

Tube length, L (m) 1,0

Fluid flow, Q (mL/min) 10

Fluid specific mass, ρβ (kg/m³) 1000

Fluid viscosity, µβ (cP) 1,0

Reynolds number, Reβ 4,2x102

Particle diameter, dp (µm) 70

Solid specific mass, ρp (kg/m³) 4500

The boundary conditions are set so that water enters the

domain with uniform velocity, developing along the

tube length, with the no-slip condition throughout the

tube walls. Particles are injected from a surface

injection positioned in the fully developed flow region.

The DDPM-DEM requires a number of parameters to

be set up, which are listed on Table 3. A soft-sphere

approach is used to determine the deformation and

elasticity of the contact between two particles, given by

Eq. (12).

Table 3. DDPM-DEM parameters.

Fluid inlet velocity, uβ (m/s) 0,85

Particle injection velocity 1,7

Particle injection points 9

Particle flow (part/s) 18x103

Spring coefficient, kn (N/m) 30

Particle-particle restitution

coefficient, ep-p 0,75

Particle-wall restitution

coefficient, ep-w 0,75

Fluid time step, Δtβ (s) 5x10-4

Particle time step, Δtp (s) 5x10-6

For the DTB simulations, the domain is discretized with

mesh elements able to compute the laminar flow and

boundary layer effects with less than 10% deviation

from the analytical solution for the velocity and

pressure profiles. The cross-section and the axial-

section view of the mesh with 6,2x104 elements is

shown in Figure 5. The particles are injected through

nine injections points, tending to settle down in the tube

wall, and being dragged by the flow toward the pressure

outlet. The main response variable of DTB

experimental test is the pressure differential along the

pipe, as presented in Figure 5. The pressure increases

due the action of the particles, stabilizing after the solid

phase starts leaving the domain and the particle number

stays approximately constant within the geometry.

Therefore, the flow is sensible to the particle’s effect,

elucidating the potential of the DDPM-DEM to take

into account the effects of the particle adhesion and

growth, to be implemented in the sequence.

Figure 5. Results for solid-liquid numerical solution with

pressure response from the fluid.

The growth phenomenon is represented through

diameter variations, which are implemented via user

defined functions (UDF), coupled with the DDPM-

DEM model. The expressions programmed in the

UDFs correlate the diameter value with variables such

as time, space and material properties. The diameter

values varied from 30 to 70 µm, with collision

parameters remaining the same as Table 3. Results for

the DTB including the effects of diameter growth are

presented in Figure 6 by different growing patterns

used for the functions.

In the simulation with diameter growing with flow

time, shown in Figure 6 (a), all particles grow

uniformly, while in the results showed in Figure 6 (b),

particles grow as their residence time in the domain

increases, achieving their maximum size at a larger

distance from the entrance. The diameter variation with

the longitudinal position, depicted in Figure 6 (c) ,

displays similar behavior as in Figure 6 (b), although

their growth is related to their distance from the inlet.

The diameter function (d) and the longitudinal and

radial growth diameter function (e) are both evaluated

with nine injection points to better illustrate size

differences. The particles grow when they move closer

to the outlet and to the walls. The expression is written

so that they shrink if they move toward the tube center,

indicating the possibility of diameter change in both

ways. The radial and longitudinal simulation also

demonstrates the possibility of using several variables

in a same function. The DDPM-DEM model is versatile

when responding to different diameter functions with

the flow reacting accordingly to the particle size

changes, as shown by the gravity influence on heavier

particles.



Figure 6. Simulation of the liquid-solid flow with diameter

varying with: (a) Flow time; (b) Residence time; (c)

Longitudinal position; (d) Radial position and (e)

Longitudinal and radial position.

The particulate flow with the effect of diameter growth

is simulated in a heterogeneous porous medium

conceived as a staggered array of cylinders. In order to

observe the particle retention, the array has an

anisotropic and an isotropic region regarding the

variation of the porosity, φ [-], and the pore throat, pt

[mm], minimum distance between adjacent cylinders,

as depicted in Figure 7. The particles enter the domain

with a diameter smaller then the minimum pore throat

size, passing through without plugging the media. The

UDF is programmed with a residence time function that

activates after 5 seconds of particle injections, making

the diameter change from 0,5 to 1,0 mm, blocking the

smaller pore regions. The problem parameters are

detailed in Table 4.

Figure 7. Porosity and pore throat spatial variation for the

staggered array of cylinders.

Table 4. Porous media geometry and simulation parameters.

Porous media dimensions (mm) 180 x 90

Anisotropic region length (mm) 90

Isotropic region length (mm) 90

Pore throat (mm) 2,4-0,6

Fluid density, ρβ (kg/m³) 1181

Fluid viscosity, µβ (Pa·s) 0,0195

Flow Reynolds number, Reβ 100

Particle diameter range, dp (mm) 0,5-1,0

Particle density, ρp (kg/m³) 1181

Spring coefficient, kn (N/m) 80

Particle-particle restitution

coefficient, ep-p 0,90



Fluid time step, Δtβ (s) 1x10-2


The particles in the domain just before the diameter

change starts are shown in Figure 8 (a) and after 3

seconds of growth in Figure 8 (b). Particles near the

outlet are trapped, because the pore throat is lower than

the respective diameter. This shows an appropriate

response from the DDPM-DEM, which is able to

acknowledge the growth of an agglomerate of particles

near a wall, respecting the boundary conditions set.

Figure 8. Particle growth in an anisotropic array of

staggered cylinders.

The adhesion model is tested with the software Rocky

DEM, which already has implemented the adhesion

equations. The variation of the adhesion distance is



analyzed by verifying the rest angle of an agglomerate

of particles in a vertical tube that lifts up. The particles

then arrange themselves in a horizontal flat tray, with

the angle formed by the pile measured. The parameters

for the simulation are listed on Table 5. It is relevant to

notice that although the spring-dashpot collision force

model is also used, as stated in Eq. (12), Rocky DEM

utilizes the Young Modulus as input for the spring

coefficient calculation.

Table 5. Adhesion force model test parameters.

Tube diameter, D (m) 0,1

Tube length, L (m) 0,7

Particle diameter, dp (m) 0,025

Particle density, ρp

(kg/m³) 1650

Particle-particle

restitution coefficient, ep-

p

0,30



Particle-particle friction

coefficient, µp-p 0,7

Particle-wall friction

coefficient, µp-w 0,5

Young Modulus (N/m²) 1x108

Particles injected 6000


In the constant force adhesion model described in

Section 2 it is necessary to specify the adhesive force

intensity as well as the sphere of influence of the

adhesion around the particles. Figure 9 shows the

results varying the minimum adhesion distance from

zero to 1x10-3 m. The increase in this value produced in

a larger portion of particles subjected to the constant

adhesion force. As a result, the particles became more

agglomerated, culminating in adhesion forces strong

enough to hold them inside the vertical tube.

Figure 9. Rest angle test results for different adhesion

distances.

The adhesion model is also tested in the Dynamic Tube

Blocking Test geometry, with the coupling of the

softwares ANSYS Fluent and Rocky DEM. The first

simulates de fluid flow phase, using the parameters

listed in Table 2, while the second calculates the solid

phase with the adhesion effects included. The solutions

communicate and influence one another.

The parameters for the particles are listed in Table 6,

with the results shown in Figure 10.

Table 6. Solid phase parameters for adhesion simulations in

DTB geometry.

Particle diameter, dp

(µm) 25

Particle density, ρp

(kg/m³) 4500

Particle-particle

restitution coefficient, ep-

p

0,3



Particle-particle friction

coefficient, µp-p 0,3

Particle-wall friction

coefficient, µp-w 0,3

Young Modulus (N/m²) 1x106

Particle flow (part/s) 18x103


Figure 10. Results for the constant adhesion force model in

a capillary tube.

The necessary force for particles to visibly adhere to the

wall was set as 250 times their weight. Even though the

adhesion criteria is met near the particle injection

surface, the flow pushes the attached solids along the

wall length and produce a thin particle layer that

spreads through the tube instead of accumulating near

the entrance, revealing a great impact of the fluid phase

in this phenomenon, even with the great magnitude of

the adhesion force set.

4. Conclusions.

Inorganic scaling formation and deposition in oilfield

downhole equipment and completion systems can

provoke serious issues ranging from productivity

reduction to increased corrosion rates and equipment

damage. A numerical model to simulate the scaling

process as a solid-liquid flow, including the effects of

crystal growth and adhesion is proposed, an

advantageous alternative to expensive and limited



experimental tests and a complement to the several

thermodynamic models available, which do not include

relevant hydrodynamic effects in their calculations.

The method proposed uses an Euler-Lagrange approach

with the DDPM-DEM model with four-way coupling.

The model showed its capacity of perceiving the solid

phase influence on the fluid phase through the pressure

differential increase following the injection of solid

particles in a fully develop fluid flow. Crystal growth is

represented through User Defined Functions with

expressions for the particles diameter. Results for the

simulation in the DTB geometry (capillary tube) and in

a heterogeneous porous media demonstrated the

versatility of UDFs and the ability of the method in

adjusting particle forces calculations according to the

size of the spheres and respecting boundary conditions.

Finally, the adhesion mechanism is evaluated through a

rest angle test, elucidating the influence of the constant

adhesion force input parameters into the pile

morphology. The adhesion forces into the liquid-solid

flow are testes in the DTB geometry, with results

revealing a significant influence of the flow in the

particles deposition throughout the geometry, which

needed a great adhesion force to compensate for this

effect. Further study calibration of the DDPM-DEM

applied for the DTB problem will refine the model’s

capacity of simulating the solid-liquid flow with an

accurate representation of the growth and adhesion

effects of the scaling process.

5. Acknowledgments

The authors are grateful to the Brazilian Petroleum

Agency (ANP), the Human Resources Program for the

Petroleum and Gas Sector PRH-ANP (PRH10 –

UTFPR) and CENPES-PETROBRAS for the provided

financial support.

6. References [1] A. A. Olajire, “A review of oilfield scale

management technology for oil and gas

production,” J. Pet. Sci. Eng., vol. 135, pp.

723–737, 2015.

[2] M. S. Kamal, I. Hussein, M. Mahmoud, A. S.

Sultan, and M. A. S. Saad, “Oilfield scale

formation and chemical removal: A review,”

J. Pet. Sci. Eng., vol. 171, no. January, pp.

127–139, 2018.

[3] A. B. BinMerdhah, A. A. M. Yassin, and M.

A. Muherei, “Laboratory and prediction of

barium sulfate scaling at high-barium

formation water,” J. Pet. Sci. Eng., vol. 70, no.

1–2, pp. 79–88, 2010.

[4] C. Fan, A. Kan, P. Zhang, and M. Tomson,

“Barite Nucleation and Inhibition at 0 to

200°C With and Without Thermodynamic

Hydrate Inhibitors,” SPE J., vol. 16, no. 2, pp.

20–22, 2011.

[5] S. J. Dyer and G. M. Graham, “The effect of

temperature and pressure on oilfield scale

formation,” J. Pet. Sci. Eng., vol. 35, no. 1–2,

pp. 95–107, 2002.

[6] M. F. Da Silva, C. C. Jacinto, C. Hernalsteens,

M. Nobrega, M. Defilippo Soares, E.

Schnitzler, and L. B. Nardi, “Cableless

Intelligent Well Completion Development

Based on Reliability,” Offshore Tecnhology

Conf., no. July, pp. 1–10, 2017.

[7] B. Popoff and M. Braun, “A Lagrangian

Approach to Dense Particulate Flows,” in

Proceedings of the ICMF 2007, 6th

International Conference on Multiphase Flow,

2007.

[8] P. A. Cundall and O. D. L. Strack, “A discrete

numerical model for granular assemblies,”

Géotechnique, vol. 29, no. 1, pp. 47–65, 1979.

[9] N. Mordant and J. Pinton, “Velocity

measurement of a settling sphere,” Eur. Phys.

J., vol. 352, pp. 343–352, 2000.

[10] P. Gondret, M. Lance, and L. Petit, “Bouncing

motion of spherical particles in fluids,” Phys.

Fluids, vol. 14, no. 2, pp. 643–652, 2002.

[11] H. F. L. Santos, B. B. Castro, M. Bloch, A. L.

Martins, H. E. P. Schlüter, M. F. S. Júnior, C.

M. C. Jacinto, and F. F. Rosário, “A Physical

Model for Scale Growth during the Dynamic

Tube Blocking Test,” OTC Bras., no. Figure 1,

pp. 1–20, 2017.

euler-lagrange numerical model for the sulfate scale ...enahpe2019.ipt.br/arquivos anais do...

Documents