Non-Newtonian behavior of a Colombian heavy

crude oil: creep and interfacial rheology tests

Katherine Ardila Morales

ABSTRACT

Flow assurance in heavy crude oil is a challenge for the oil industry, given its high viscosity

and low mobility. The present study shows the non-Newtonian behavior exhibited by a

13ºAPI heavy crude oil when performing flow, oscillatory, and creep experiments. Flow

testing shows increases in thixotropy as periods of stress and rest accumulate over 30 days.

These changes are measured with the thixotropic area of the loop formed by the decay and

recovery of viscosity. Oscillatory tests reveal the reversible nature of these viscosity changes

since the bulk moduli do not change between loop tests. The viscoelastic effects of stress

history are observed in the early stages of creep tests. These show a series of instantaneous

elastic deformations that are understood as the solid-liquid transition in percolated structures.

Interfacial shear rheology experiments with a surface coverage of 1,5 𝑚𝑔

𝑚2 show a structure

that flows like a soft glass. The reproducibility of the linear viscoelastic zone of an interface

with stress history provide evidence that the changes between shear-induced metastable

states are reversible. The absence of hysteresis in flow tests probes that changes of structures

with these characteristic times (0.2 s) are negligible in flow tests. The results suggest a

connection between the thixotropy and the viscoelasticity of the crude oil, defined by the

breakdown and reconstruction of the structures formed by the asphaltenes whose elastic

contribution is visible only at low shear stress.

Keywords: Thixotropy, heavy crude oil, non-Newtonian, asphaltenes.

2

INTRODUCTION

In the current context of global energy security, it is irrefutable the fact that the highest

percentage of energy demand is covered by fossil fuels1. As conventional reserves are

depleted, the production of heavy oil and other unconventional resources must be improved.

There are proven reserves of heavy and extra heavy crude oil that could cover 90% of the

demand for fossil fuels by 21002. However, in many cases, the production of heavy crude oil

is considered unprofitable, given the multiple operational problems that they represent.

Heavy crudes oils are classified as hydrocarbons between 10-22.3ºAPI gravity, low H/C

ratio, high viscosity, and chemical complexity due to the presence of asphaltenes, resins,

sulfides, metals, and heteroatoms3. Their high viscosity and low mobility represent a high

cost in production facilities and pipeline transport.

Flow assurance of heavy crude oil is a challenge for the oil industry because large amounts

of energy are necessary to achieve a pressure difference to effectively transport the crude oil,

especially after shut-down operations, when the viscosity of the crude oil increases.

Providing the energy for start-up operations is not the only problem; When the heavy crude

oil flows through a pipe, the liquid near the pipe wall is subjected to the highest shear rate

and the lowest velocity; resulting in a stronger breakdown that leads to lower viscosity near

the wall, causing a lubricating effect for the fluid in the middle of the pipe. This type of

annular flow generates a non-linear pressure profile. It allows asphaltene precipitation and

blockages at low pressure points and makes it difficult to characterize the lubricating layer

generated by non-newtonian behaviors as thixotropy4.

Initially, shear thinning behavior is attributed to viscous heating by Newtonian and non-

Newtonian fluids which magnitude increases at high Nahme numbers Na (high shear rate)5.

However, typical rheological behavior, such as the increase in viscosity with rest time and

apparent yield stress, led to research on the non-Newtonian behavior of heavy crude oil,

focusing on viscoelasticity and viscosity dependence on the type of flow and measurement.

An important physical phenomena associated to heavy and extra heavy crude oils is

thixotropy. It is defined as the decrease in apparent viscosity when the sample flows under a

3

constant shear rate or shear stress. When shear stress or shear rate is removed, the fluid

gradually recovers until it reaches initial viscosity. The effect is time dependent6.

This reversible change in viscosity is due to viscous heating and to the rupture and subsequent

reconstruction of the microstructure. The fully developed structure at rest periods gives the

fluid its zero shear viscosity. When energy is added to the system in the form of shear stress

(i.e. 𝑃𝑒 ≫ 1), the structure changes in two different ways; it is fragmented due to the

hydrodynamic forces acting on it, and new structures of smaller volume are formed due to

increased number of collisions. The shear thinning is evidence of this breakdown. When the

flow stops (i.e. 𝑃𝑒 ≪ 1), the structure is rebuilt due to Brownian effects7. The higher the

difference between the times of breakdown and buildup, the higher the grade of thixotropy6.

Thixotropy in heavy oil could be caused either by maltenes or asphaltenes8,9. The last ones

are defined as the most polar fraction the crude oil, not soluble in straight-chain hydrocarbons

as pentane or heptane. Although they do not have a unique molecular structure, it is known

that they are the components with the highest molecular weight and with the most significant

presence of heteroatoms and metals. Asphaltene form clusters that co-exist with other solids

(like resins and clays) in a colloidal suspension. In the presence of water, they can act as

surfactants, stabilizing petroleum emulsions. The molecular structure of asphaltenes gives

them a particular self-association nature10–12. This tendency is the reason why they are

considered as the component capable of forming networks or aggregates that break and

rebuild under different flow conditions.

The starting point of this study is hysteresis loop testing as a practical alternative to observe

the thixotropic behavior of heavy oil5. Previous studies have shown that thixotropy grade is

measured by the changes in the hysteresis area when the test is performed after large

deformation and rest stages13–16. In this study, oscillatory tests were introduced at different

times of a proposed flow protocol with large deformation and rest stages to observe the

viscoelastic behavior and its relation with hysteresis area. Further viscoelastic measurements

were obtained from creep tests. This protocol consisted of overlapping creep tests of different

stress and the addition of dynamic and static (zero-shear) recovery stages. These results were

related to additional flow tests for lower stress values

4

Finally, the viscoelastic properties and the viscosity of structures formed by asphaltenes were

studied using interfacial shear rheology. Results were useful to link the molecular behavior

of a structure to the macroscopic behavior of heavy crude oil.

Materials and Methods

Crude oil

A sample of 1000 mL of Colombian heavy crude oil was dehydrated and stored in a

hermetically sealed glass container used for all the experiments. The container was left

undisturbed in a dark cabinet at ambient temperature (18°𝐶). Characterization of crude oil is

presented in Table 1.

Table 1. Characterization of crude oil used in this work

Parameter Method Value

Saturates (wt %)

IP 469 17

7.4

Aromatics (wt %) 37.8

Resins (wt %) 15.3

Asphaltenes (wt %) 39.5

Density (kg/m3) at 15.5°C ASTM D7042-19 18 954

Density (°API) at 15.5°C ASTM D7042-19 18 13.6

Basic Sediment and Water (vol %) ASTM D4007-11(2016)e119 0.2

Solvents and chemicals

N-heptane (HPLC grade, ≥ 99%), for asphaltene precipitation, and N-dodecane (anhydrous,

≥ 99%), used as oil phase in interfacial experiments, were purchased from Sigma-Aldrich

Co. Chloroform (HPLC grade, ≥ 99%), used for all asphaltene solutions. Ultra-pure water

with a resistivity of 18.5 MΩ.cm (pH 7.0) was used as sub-phase for interfacial experiments.

Asphaltene compound

Asphaltenes were extracted from the Colombian crude oil mentioned following ASTM

D3279-12 procedure20. Crude oil was dissolved in a 1:40 g/g ratio with n-heptane. This

solution was gently stirred using magnetic stirring for 24 hours at 22 ºC. The solution is then

vacuum filtered using a 2.5 µm microfiber filter. The filter cake is allowed to dry out under

a confined nitrogen atmosphere for 36 hours in order to reduce n-heptane content. An

5

asphaltene solution in chloroform was prepared at 0.1 g/L concentration. The same batch of

solution was used for all the interfacial experiments; it was sonicated for 5 minutes before

interface loading and refrigerated in between tests to avoid oxidative aging21.

Bulk rheological measurements

All bulk rheological experiments were performed at atmospheric pressure using a controlled-

stress rheometer (AR G2, TA Instruments) equipped with a Peltier cooling/heating system

and with a concentric cylinders Couette geometry of 5 mm gap to ensure minimal damage in

sample placement and effectiveness at large shear rates or large amplitudes of oscillation22,

23. The total amount of sample loaded was 20 mL per experiment.

Flow sweep experiments

The sample was conditioned at a temperature of 25 ºC and pre-sheared at 500 𝑠−1 for 5

minutes. First shear rate sweep was performed from 1 𝑠−1 to 3000 𝑠−1, followed by a second

shear rate sweep from 3000 𝑠−1 s to 1 𝑠−1. Measurements were taken using 5 points per

decade.

Shear peak hold experiments

This test purpose was to exert a prolonged deformation by holding a constant shear rate of

1000 𝑠−1 for 4 hours to a sample conditioned at 25 ºC. Peak-hold experiments were

performed as intermediate stages between flow sweeps.

Oscillatory tests

Amplitude sweeps were performed from 1% to 600% oscillatory strain, and frequency

sweeps from 0.0001 rad/s to 500 rad/s to obtain elastic and viscous moduli behavior of the

system. Experiments were conducted at 25ºC. Measurements were registered using 5 points

per decade.

Creep experiments

Creep tests consist of imposing a constant stress on the sample for a certain period until

steady-state is reached. In this case, steady state was established as 0.1% change in the strain-

time slope. Multiple creep stages were performed with shear stresses of 2 Pa and 20 Pa with

6

dynamic and static recovery stages. Creep tests had an average duration of 200 s. The

recovery lasted 10 minutes in both cases, dynamic and static.

Temperature ramp test

The sample was kept under quiescent conditions for 20 minutes at the initial test temperature

before starting the test. Then, a constant shear rate of 1 𝑠−1 was held while the temperature

changed from 30 ºC to -5 ºC.

Interfacial rheological measurements

A double wall ring (DWR)24 geometry in an imposed stress rheometer (AR G2, TA

Intruments) was used to characterize the shear interfacial behavior of asphaltenes at the

dodecane-water interface. Asphaltenes solutions were carefully deposited dropwise between

the two phases using a 1μL gas-tight syringe (Hamilton Company, USA). The amount added

corresponded to a surface coverage of 1,5 𝑚𝑔

𝑚2 distributed on the interfacial area restricted by

the sample holder and the ring. The sample holder had a 6,13𝑥10−4 𝑚2 area between the

ring and the internal wall and a 9,42𝑥10−4 𝑚2area between the ring and the external wall.

The sample was left for 30 minutes to ensure solvent evaporation. All procedures were

performed at 25 ºC and atmospheric pressure.

Interfacial flow sweep experiments

This test was divided into two parts; the first is a shear rate flow ramp from 0,001 𝑠−1 to

10 𝑠−1; and the second, a reverse flow ramp from 10 𝑠−1 to 0,001 𝑠−1. Shear rates used for

interfacial experiments are usually lower than those used for bulk tests to avoid interfacial

rupture17. Each flow ramp lasted 30 minutes, with 81 points taken.

Interfacial oscillatory tests

An amplitude sweep from 0,1 % to 100 % strain was conducted to establish linear

viscoelastic zone. A frequency sweep was then performed in the range of 0,01 𝑟𝑎𝑑

𝑠 to 100

𝑟𝑎𝑑

𝑠

at constant strain of 10 %.

7

Results and discussion

Flow sweep and peak-hold experiments

The idea of using the loop formed by two flow tests first arose in 1946 as a way to identify

the effect that a progressive increase and subsequent decrease in the deformation rate had on

the viscosity of a fluid25. The difference between two flow curves forming a loop is known

as hysteresis and, when referring to thixotropic behavior, this hysteresis is attributed to

internal changes in the microstructure of the fluid being evaluated14,23.

Fig. 1 Flow sweeps of heavy crude oil measured at 25℃ before and after induced shear history

(marked as “post peak hold”, 1000 𝑠−1 for 4 h) forming two loops, the second exhibits higher

hysteresis. All curves correspond to flow sweeps going from 1𝑠−1 to 3500𝑠−1 and back from

3000𝑠−1 to 1𝑠−1.

Fig. 1 presents the initial loop formed by two stages of flow after a 5-minute conditioning

pre-shear step; the first, generated by a stepped increase in shear rate, shows viscosity values

higher than the second, with a decreasing shear rate. Following the first loop test, the sample

was subject of a 4-hour peak-hold stage to induce prolonged deformation over the sample to

assess possible long-term effects. Viscosity did not exhibit significant variations in the peak-

hold test compared to flow sweep tests. Thixotropy can be analyzed by the size (area between

two flow sweeps) and shape of the hysteresis loop. Higher hysteresis areas correspond to a

stronger degree of thixotropy, and pure thixotropy is observed when changes in viscosity are

1

10

1 10 100 1000 10000

Vis

cosi

ty

[Pa.s

]

Shear rate [1/s]

Initial

Post Peak Hold

8

reversible (i.e. closed loops); otherwise, time dependency is influenced by viscoelastic

contributions26. According to the size and shape of both loops in Fig. 1, it could be concluded

that crude oil exhibits pure thixotropy, and it is increasing over time.

Studies on colloidal suspensions attribute the difference between the path of the two curves

to internal structure breakdown and buildup phenomena27. The breakdown stage occurs when

the shear rate increases, causing the viscosity to decrease, and the buildup stage of the

structure occurs when shear rate decreases (including rest conditions), recovering initial

viscosity values. Physical causes of irreversible changes include strong orthokinetic

aggregation and are usually labelled as aging26.

Fig. 2 Flow sweeps of heavy crude oil measured at 25℃ in between rest periods of 12 hours, 15

days and 30 days showing changes in hysteresis loop size and shape. All flow sweeps were

perfomed from 1𝑠−1 to 3000𝑠−1 and back. There was a 4-hour peak hold stage after initial loop.

The influence of resting periods over breakdown/buildup paths was assessed through a 30-

day procedure. Loops hysteresis tests performed in this procedure are shown in Fig. 2.

Hysteresis areas and reversibility do not show mayor changes after 12 hours and 15 days of

rest. On day 30, there is an increase in the thixotropic area and irreversible change in viscosity

(i.e. open loop). It is possible that this behavior is the result of a change in the viscoelastic

nature of the system6 or an indication of aging after multiple buildup/breakdown stages 28–31.

1

10

1 10 100 1000 10000

Vis

cosi

ty

[Pa.s

]

Shear rate [1/s]

Initial

12 h

15 days

30 days

9

From Fig. 2, it is also important to mention that the viscosity of the fluid increases with time

(until reaching a difference of 5 Pa.s on day 30 compared to the initial sample). This increase

is due to the evaporation of light components of crude oil during periods of flow, transfer and

storage of the sample.

Fig. 3 shows the influence of Light-ends evaporation on the viscosity of crude oil, where

evaporation stages were introduced before flow stages. The procedure was as follows: The

crude oil sample was heated to 75 ºC for one day in an open container, then it was allowed

to cool down until ambient temperature was reached where a second loop test at 25 ºC was

performed. The procedure was repeated with an evaporation time of 3 days, but on the third

day, the flow sweep was performed until 100𝑠−1 due to torque limit of the instrument. The

Initial loop is also shown as a reference sample with unaltered composition. Heating heavy

crude oil to 75 ºC produces the separation of low molecular weight compounds32,33, , so it

was expected that the mass percentage of asphaltene would increase with the evaporation

time. Maltenes´s mass loss was 3% for one day of evaporation and 7% for three days.

One can identify a substantial increase in viscosity as light-ends of crude oil are lost.

However, the logarithmic viscosity does not allow to see the change in the thixotropic areas,

so calculated areas related to the asphaltene mass fraction on each measurement are shown

in Table 2.

Table 2. Thixotropic area and asphaltene mass fraction after light-end evaporation from the

data of Fig. 3.

Thixotropic area

[x105 Pa/s]

Asphaltene

mass fraction

0.13 0.395

0.23 0.407

1.1 0.423

The thixotropic behavior is highly influenced by the packing factor of flocculated networks

formed by complex colloidal systems. The packing factor depends on the size and shape of

the particles capable of building structures. Spherical particles with an uneven size

distribution have higher packing factors than non-spherical particles with uniform size

10

distribution; This is because the smaller particles fill the free spaces, strengthening the

flocculated networks6.

Fig. 3 Flow sweeps of heavy crude oil measured at 25℃ in between evaporation rest periods of 1

day and 3 days. Initial and 1 day flow sweeps were performed from 1𝑠−1 to 3500𝑠−1 and back. 3

day flow sweep was performed from 1𝑠−1 to 100𝑠−1and back.

In suspensions with well-characterized components, it is possible to calculate a maximum

packing factor, and even co-relate the packing factor of a stressed fluid with the presence of

different structures that generate changes in their rheological behavior34. Estimating the

packing factor of the structures formed by asphaltenes and resins compounds in crude oil

during flow or rest is a complex process since the shape, size and aggregation dynamics of

the aggregates or networks is uncertain. However, the self-assembly nature of asphaltenes

and the mass reduction in the continuous phase (evaporated maltenes) could lead to higher

packing factors and enhance the ability to build flocculated networks. Based on Fig. 2, it

seems possible that the heavy crude oil with different stress histories reached similar packing

factors by 3000𝑠−1.

So far, loop tests have shown a drop in viscosity at around 100𝑠−1 in all scenarios, meaning

that the effects of hydrodynamic forces on structures are independent of zero-shear viscosity.

Hydrodynamic forces can change the orientation, size and density of structures. In a weakly

1

10

100

1 10 100 1000

Vis

cosi

ty [

Pa.s

]

Shear rate [1/s]

Initial

1 day

3 days

11

flocculated system, such as undisturbed crude oil, low shear rates do not generate a decrease

in viscosity as the structure reorganizes under the influence of Brownian motion, maintaining

the resistance to flow. When the shear rate is increased (hence higher Pe), structure

components are fragmented and oriented in a velocity gradient direction; at this point, the

hydrodynamic forces predominate over Brownian motion35. When shear rate is removed,

Brownian motion rearrange the structures, restoring the average viscosity to its initial value.

It could be said that the hysteresis is due to the delay in the restoration of the structures,

however, this does not explain higher delays and irreversible changes in viscosity over time

(day 30).

Regarding the size and density of the structure, it is possible to affirm that the hydrodynamic

forces promote the increase in the number of collisions between moving particles, and

considering the self-assembly nature of asphaltenes, a significant number of these collisions

are likely to be effective which results in larger and more densely packed aggregates. Further

changes in density and size are hindered due to the impossibility of more aggregation and by

attrition at higher shear rates36–39.

According to Fig. 2 at 3000𝑠−1 the heavy crude oil seemed to have reached the limit of

aggregation. By decreasing the shear rate, the structures are reorganized through diffusion to

give rise to weakly flocculated networks, which occupy a higher volume. The recovery of

the viscosity is observed in the test time, which means that the restructuring of the flocculated

network by diffusion does not require long relaxation times. The evolution of the formation

dynamics of these structures caused by the stress history (aging) could explain the increase

in the thixotropic area and the irreversibility of viscosity over time.

Temperature ramp test

In some cases, thixotropic behavior and the presence of a yield stress is attributed to wax

content in crude oil13,14,40–42. Waxes are said to be made of cyclic, branched, and long-chain

paraffins with the latter being able to form gels sensitive to temperature, even at

concentrations as low as 0.5 wt %. The usual gelation temperatures of waxy crude oils are

around 35 ºC. However, different studies have shown that in the presence of asphaltenes,

12

gelation temperature and yield stress decrease—because asphaltenes act as "weak points" in

paraffin gel-like structures43–47.

If thixotropy were the result of waxy gelled structures, there should be a gelation temperature

close to the testing temperature used until now (25 ºC), so that the flow tests were affected

by the onset of paraffin gelation process.

Fig. 4 Temperature ramp from 30 ºC to -5 ºC. The shear rate was set at 1𝑠−1 and maintained for 70 minutes.

Fig. 4 shows the evolution of the stress as a function of a temperature ramp under a constant

shear of 1𝑠−1. The sample was first heated to 50 ºC for 15 minutes to avoid any possible

thermal memory noise 46. The increase in shear stress with temperature does not show plateau

areas associated with yield stress or other disturbances that would indicate the onset of wax

gelation, so it is clear that the structures responsible for thixotropy do not correspond to a

asphaltenes disrupted wax gels.

Oscillatory and creep experiments

To determine if the ability to store and dissipate energy of asphaltene structures had an

influence on the fluid's average viscoelastic properties, the viscoelastic behavior was

analyzed from two approaches: oscillatory tests and creep tests

1

10

100

1000

-10010203040

Shea

r st

ress

[P

a]

Temperature [ºC]

13

Fig. 5 Frequency sweeps with a 100% strain of heavy crude oil measured at 25℃. Initial

measurement was performed on an undisturbed sample, the second measurement immediately after

peak-hold and third measurement after 12-hour rest.

Oscillatory tests for frequencies between 0.1-100 rad/s and strain between 0.1-200% were

performed to characterize the viscoelastic behavior of the fresh samples. The frequency

sweeps were introduced between flow and rest stages, to evaluate changes in viscoelastic

properties of crude oil induced by a prolonged deformation and a subsequent recovery. Fig.

5 shows a frequency sweep with a 100% strain carried out on the fresh sample (marked as

initial), immediately after the 4-hour peak hold at 1000𝑠−1, and after 12 hours of resting

time. The tendency of the moduli is almost the same for all test times, except for a slight

change in the storage modulus at lower frequencies (0.1 rad/s).

Table 3 shows the calculated slopes and intercepts for the trend line of moduli in the three

stages. A slope ratio of 2:1 for elastic and viscous modulus enables a fit to a Maxwell

viscoelastic liquid model for the for measurements after the peak-hold. Complex moduli 𝐺∗

between 1 and 80 Pa were calculated with the Eq. (1) for the measurements after peak-hold.

The maximum relaxation time of the crude oil was estimated by extrapolating the linear trend

line of moduli to find crossover frequency.

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

1,E+01

1,E+02

0,01 0,1 1 10 100

Moduli

[P

a]

Frequency [rad/s]

G' InitialG' Post peak-holdG' 12 hG'' InitialG'' Post peak-holdG'' 12 h

14

Table 3. Linearization of moduli.

Moduli

Slope

[Pa.s/rad]

Intercept

[Pa]

Crossover

Frequency [Pa/s]

G’ Initial 1.574 -1.540 10156.83

G’’ Initial 0.997 0.773

G’ Post peak-hold 1.977 -1.781 423.03

G’’ Post peak-hold 1.001 0.783

G’ 12 h 1.990 -1.447 387.95

G’’ 12 h 0.999 0.910

The average frequency of the transition point for measurements after peak-hold was 405

rad/s, with a deviation of 24 rad/s. By equating the elastic modulus of Eq. (2) and viscous

modulus of Eq. (3), and substituting the viscosity for time using Eq. (4), it is possible to

obtain a simple expression, Eq. (5), for the relaxation time in terms of the frequency.

𝐺∗ = √𝐺′2 + 𝐺′′2 𝐸𝑞. (1)

𝐺′ =𝐺(𝜔𝜏)2

1 + (𝜔𝜏)2 𝐸𝑞. (2) 𝐺′′ =

𝜂𝜔

1 + (𝜔𝜏)2 𝐸𝑞. (3)

𝜏 =𝜂

𝐺 𝐸𝑞. (4) 𝜏 =

1

𝜔 𝐸𝑞. (5)

The complex moduli and the maximum relaxation time are closer to those found in polymeric

solutions than other viscous oils48, suggesting the presence of a structure that provides the

elastic component, as polymers do, but may relax in shorter times than those corresponding

to the tested frequencies. According to these observations, the aggregation dynamics of the

structure that cause the thixotropic behavior of the crude oil do not seem to alter the moduli;

for the tested frequencies, stressed crude oil flows in the viscous region after shearing and

resting stages, which means that the maltenes may have a more significant contribution to

the viscoelasticity of the system9.

Nevertheless, to have a complete picture of the viscoelastic behavior of crude oil and its

response to another type of deformation, creep tests were carried out. As expected in a

15

viscous fluid, the steady-state at low shear rates was reached before 200 s for all the

experiments. Fig. 6 shows the scheme of the three sets of creep tests proposed. The objective

of the included modifications was to determine the influence of the recovery conditions on

the reorganization of the structure. Tests were performed without recovery, with recovery in

non-shear stress conditions (0 Pa) and with recovery under low stress.

Fig. 6 Scheme of three sets of creep. Set 1 consists of two consecutive creeps, 20 and 2 Pa. Set 2 includes 600 s recovery at 0 Pa between two creeps. Set 3 includes 600 s recovery at 2 Pa between

two creeps.

Fig. 7 shows the data obtained for times between 0-1 s, given the assumption of having

structures with short characteristic times. Data in Fig. 7a consisted of two consecutive creep

steps of Set 1; the first of 20 Pa (open squares) and the second of 2 Pa (closed squares) with

no intermediate recovery time. Fig. 7b shows only the second creep after recovery (after 800

seconds); the red squares show the creep after a dynamic recovery of Set 2, and the blue

squares show the creep after a static recovery of Set 3. The selected shear stress values

16

correspond to the start and the average deflection point (100𝑠−1) of the viscosity in the flow

tests.

Fig. 7 Creep experiments for 20 Pa and 2 Pa with no recovery in between (a) and second 20 Pa

creep after 10 minutes at 2 Pa, marked as dynamic recovery, and after 10 minutes of rest, marked as static recovery (b) all performed at 25 ºC.

Fig. 7a shows how the unaltered crude oil sample exhibits viscoelastic behavior without

instantaneous deformations, with a delayed elastic response, which is evident in early times,

1,E-03

1,E-02

1,E-01

1,E+00

1,E+01

1,E-03 1,E-02 1,E-01 1,E+00

Str

ain

Time [s]

a)

Creep 20 Pa

Creep 2 Pa

1,E-03

1,E-02

1,E-01

1,E+00

1,E+01

1,E-03 1,E-02 1,E-01 1,E+00

Str

ain

Time [s]

b)

Dynamic recovery

Static recovery

17

extending up to 200 seconds, when the pure viscous regime is reached. In the second, lower

stress creep, some instantaneous strain increments are observed, showing the presence of

multiple deformations of purely elastic (or plastic) character that may correspond to critical

stress points of the recovered structure after the first creep.

Instantaneous deformations not only happen when shear stress decreases. The results in Fig.

7b show instantaneous strains for a second creep of the same magnitude as the first (20 Pa)

after a 10-minute recovery. The first instantaneous strain is higher when the recovery stage

is carried out at non-flow conditions. That means there is a structure with more resistant

architecture than when it is formed under flowing conditions.

Continuing with the comparison to particle systems (suspensions) that exhibit similar

behavior49. A particular example is that of magnetorheological fluids50, whose conformation

of structures is given by the orientation in the direction of an electric field. The structures in

these fluids have short relaxation times, which provides them with an elastic response to the

applied deformation. However, when the particles' density exceeds a critical point, the

structures do not associate in the same way each time they are broken by shear, so the

deformation response is plastic.

Something similar could occur with flocculated asphaltene networks. In this case, the

instantaneous deformations could demonstrate the presence of solid-like metastable

structures with short characteristic times that change their configuration with the applied

stress and with the recovery time. The average properties of all states of structure lead to

similar bulk behavior, as in the storage moduli case.

Additional flow tests were performed to see if the presence of these instantaneous

deformation points had any influence on the viscosity at less than 2 Pa shear stress values.

The procedure consisted of introducing shear stress sweeps after four previously described

experiments: the initial loop, the peak-hold, 12 hours of rest, and the loop that follows this

rest.

18

Fig. 8 shows a bifurcation of the viscosity as shear stress tends to zero; for measurements

performed before 12-hour rest, the viscosity decreases with stress, and for measurements

performed after de rest viscosity, there is an asymptotic increase. This bifurcation effect has

been previously shown and thought to be related to the presence of a yield stress13,30,40,49,51.

However, it would not be adequate to say that this particular crude oil exhibits a yield stress

because it can flow near to zero shear stress in certain steps. What can be stated, relating this

results with creep results, is that there are critical stress points at which the structure has

multiple solid-like deformations (elastic or plastic) and that these deformations influence the

viscosity of the system at lower shear rates.

Fig. 8 Flow shear stress sweeps from 1 × 10−4 𝑃𝑎 to 1 𝑃𝑎 after initial loop (open black squares),

after peak-hold (closed black squares), after 12 hours of rest (open red squares) and after a second loop (closed red squares). All tests were performed at 25 ºC without preshear.

The change in creep and bifurcation of viscosity can be related by using a simple mechanical

model. The creep test performed on the unaltered crude oil sample (open squares, Fig. 7a)

fits the Burgers model described by Eq. 6, without the first elastic term, since the

instantaneous strain is negligible. Differentiating Eq. 6 with respect to time, we obtain an

expression for the shear rate, Eq. 7.

1

10

100

1,00E-04 1,00E-03 1,00E-02 1,00E-01 1,00E+00

Vis

cosi

ty [

Pa.s

]

Shear stress [Pa]

Initial

Post peak-hold

12 hours

Post loop

19

𝛾(𝑡)

𝜎=

1

𝐺1+

1

𝐺2(1 − 𝑒

−𝑡𝜏2⁄ ) +

1

𝐺3(1 − 𝑒

−𝑡𝜏3⁄ ) + (… ) … +

𝑡

𝜂 𝐸𝑞. (6)

𝑑𝛾

𝑑𝑡= 𝜎 [

𝑒−𝑡

𝜏2⁄

𝐺2𝜏2+

𝑒−𝑡

𝜏3⁄

𝐺2𝜏3+ (… ) … +

1

𝜂 ] 𝐸𝑞. (7)

A solid-like instantaneous deformation (step in creep) is defined by a constant shear rate of

values close to zero for a finite test time. For the shear rate to be close to zero, that is, to

observe a solid strain step, two events must take place. First, the viscosity should tend to

infinity, as is the case of red data in Fig. 8, so that the viscous term is canceled out from Eq.

7. Second, the characteristic time of delayed elastic response (τ) must be short. In the case

of crude oil, the longest relaxation time is an order of magnitude less than the test time of the

first step.

A step in a creep test produced by constant low shear rate or viscosity bifurcation in flow

sweeps provide evidence of a possible transition between solid-like to a viscoelastic liquid.

The Burgers model works for the limit of this transition in the viscoelastic liquid. It is

necessary to consider a step function to analyze the rheological behavior at times close to

zero, with elastic terms at early times and viscoelastic terms (like Burgers) when the

deformation is continuous and differentiable over time.

The abrupt solid-liquid transitions of three-dimensional structures can be explained using

percolation theory. Percolation theory for solids relates structures in two and three

dimensions formed by “nodes” connected by “bridges” (i.e. bond percolation). The approach

is based on the probability of finding connected nodes or empty spaces in a graph (usually a

lattice), that is, clusters or fractures. The system is said to be percolated if there is a

continuous connection along the network from one side of the lattice to the other. In a 3D

system, the boundaries of the vessel would be considered as the lattice.

It is necessary to have a reference value to define the architecture of the solid structure at a

certain point in time, known as the critical probability Pc. If the analysis is oriented to the

bridges, Pc is the probability of finding fractures across the entire structure. If the calculated

probability is below Pc, there is a connected (percolated) structure, and for probabilities

20

above Pc nodes do not connect. If the system is percolated, it could be treated as a sol-gel,

where rheological properties such as creep strain show a sharp increase. Transitions in

percolated structures are fast, so the change from connected to disconnected nodes occurs in

a very short probability range. Graphically, the transition is described as a sea of islands

(clusters of connected nodes) swimming in empty spaces (disconnected nodes)52.

The difficulty in calculating the probabilities lies on two essential factors when choosing the

correct assembly: the number of nodes and the degree of each node, that is, its ability to link

with other nodes53. The calculation of the probability in a two-dimensional lattice of degree

4 (square) is simple, it can be solved analytically, but when the dimensions and the degree

increases, it is necessary to increase the number of nodes to have a representative system of

the real structure.

Fig. 9 A two-dimensional random geometric graph with 500 nodes and degree 5 (a) and a cubic

lattice of a gel with Pc of 0.325 (b). Images reproduced and modified from Cohen & Havlin53 and Otsubo et al54.

Fig. 9 shows two different cases of percolated networks that were built for polymers and

which architecture and solving methods may be useful to study asphaltene networks. Fig. 9a

shows a polymer structure in two dimensions, of degree 5 with nodes distributed in a random

geometry, and that appears to be in a transition stage. On the other hand, Fig. 9b shows a

three-dimensional gel structure of the cubic lattice type (grade 6) with fractures and critical

probability 0.325 calculated by the Monte Carlo method.

a b

21

In the construction of a simple structural model, one could ignore the effects of other

suspended solids that could interact with the asphaltene network structure and consider

asphaltenes as nodes and their attractive interactions as bridges, forming flocculated

networks with characteristics of fractals typical of asphaltenes10,12,55,56 with a minimum of

375 nodes57. The difficulty in building such model for asphaltenes would be in calculating

the critical probability since the characteristic self-assembly of asphaltenes is governed by

different mechanisms, some more likely to dominate than others58–60.This would make some

of the bridges between nodes more likely to bond than others, hindering the definition of a

unique grade for the nodes.

Furthermore, there is a probability that in some geometric configurations, the bonding

capacity of a node not only depends on its current degree but also decays with time or with

stress history. Hydrodynamic forces influence the critical probability of percolated networks

built by particles with weak attractive forces, enhancing the motion of colloids inside

aggregates and changing architecture to more elongated shapes61. This is what must happen

so that the aggregates that form under the shear have a limited size. This evolution of the

degree of the nodes allows representing phenomena such as aging, shear rejuvenation and

irreversible change in structure.

Various attempts have been made to mathematically describe the thixotropic behavior of

fluids62. The proposed models have three different approaches: some of them relate

thixotropy to shear rate, others to shear stress and others to the microstructure kinetics.

Remarkable advances have been made, such as the separation of the kinetics of the

breakdown and buildup stages63, the inclusion of Brownian motion64, and connections with

viscoelasticity65–67.

It is not easy to find a model that describes the rheological behavior of all thixotropic fluids,

first because most models require many input parameters and some of them are not always

possible to obtain, such as the infinite viscosity, and second because the type of structure

varies from one fluid to another and the structure parameter of most models (λ) does not

always offer enough information about the dynamic behavior of the structure.

22

Percolation theory predicts sol-gel phase transition regimes that happen in short times;

therefore, it helps to build a complete map of the structure geometry and distribution in time.

The inclusion of concepts such as critical probability in the λ structure factor might be useful

to find the average energy state of multiple structures defining the behavior of structured

fluids that cannot be described by existing models.

Interfacial experiments

The characterization of structures in three dimensions is a complex process due to the

molecular variability of asphaltene compounds and their aggregation dynamics. Interfacial

rheology is a tool used to determine the properties of two-dimensional structures, providing

useful information on the interaction of asphaltenes that can be correlated with 3-dimensional

phenomena.

All experiments were performed at a surface coverage of 1,5 𝑚𝑔

𝑚2, corresponding to the

transition between the expanded-liquid and condensed-liquid in surface pressure tests for

decane-water and air-water interfaces68. This concentration was selected to prevent the

presence of solid-like aggregates from the beginning of the test since its purpose was to

observe the influence of the shear rate on the liquid-solid transition.

The results of the oscillatory tests in Fig. 10 show that there is a linear viscoelastic plateau

for asphaltenes with an elastic modulus of 2.8 × 10−5𝑃𝑎, significantly lower than elastic

modulus reported for rigid asphaltene films under similar frequency and strain conditions29,69.

It has been proposed that when asphaltene concentration is close to a critical point, they

behave like soft glasses70. There seems to be a transition to a glassy rheological behavior at

5 rad/s, corresponding to a longest relaxation time around 2 × 10−1 𝑠, two orders of

magnitude higher than the estimated time for crude oil and, interestingly, close to the creep

test time when the instantaneous deformation steps disappear.

23

Fig. 10 Amplitude sweep at a constant frequency of 0.1 rad/s (a) and frequency sweep at a constant

strain of 10% (b). Both experiments were performed for asphaltenes at dodecane-water interface at 25 ºC.

If the viscoelastic asphaltene structure changes its configuration under stress, these changes

should be reflected in the viscosity of the interface. Fig. 11 shows the viscosity response to

an increase and subsequent decrease in the shear rate.

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

0,1 1 10 100

Modu

li [

N.m

]

Strain [%]

a

G' G''

-10

0

10

20

30

40

50

60

70

80

90

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E-02 1,E+00 1,E+02

Phase

angle

, δ

[º]

Moduli

[N

.m]

ω [rad/s]

b

G' G''

Tan (δ) δ

24

Fig. 11 Flow sweeps of asphaltenes from 1 × 10−3𝑠−1 to 10 𝑠−1 and back with no resting time in

between.

The decrease of interfacial viscosity by three orders of magnitude is evidence of the

breakdown of the initial structure. However, breakdown and buildup of the structure do not

come out the same way for asphaltenes at an interface, with 2D deformation, and asphaltenes

as part of a colloidal system with deformation in 3D. The most evident difference between

the two types of deformation is the lack of hysteresis, which suggests that the kinetics of the

processes involved in breaking and buildup are similar. There is no delay in buildup as in the

bulk phase. The second difference is the shape of the viscosity drop, which exhibits higher

slopes from the first point of shear rate, unlike the Newtonian flow plateau observed in the

bulk phase.

The structure changes are visible from the lower limit of deformation in interfacial

measurements, while the structural changes in the bulk were observed around 100 𝑠−1. The

third difference is the behavior in the upper limit of shear rate; in bulk tests, a constant

viscosity value was not achieved, and in interface tests, constant viscosity is reached at a

shear rate of 2 𝑠−1. Since the viscosity returns to its initial value, it could be said that the

change in structure is reversible

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

0,001 0,01 0,1 1 10

Vis

cosi

ty [

Pa.

s.m

]

Shear rate [1/s]

Sweep 1

Sweep 2

25

Fig. 12 Frequency sweep at a constant strain of 10% performed 5 minutes after forward and backward flow sweeps.

After the flow tests, oscillatory tests were performed to obtain more information about the

energy state of the structure after deformation. Fig. 12 shows the elastic and viscous moduli

measured after the two flow stages described above and five minutes of rest. There is an

almost imperceptible increase in the elastic modulus after flow while the viscous modulus

and the transition frequency remain the same. It is fair to say that the change in viscoelasticity

induced by the flow is negligible at surface coverage of liquid-solid transition.

The results of the interfacial experiments can be explained by the soft-glassy rheology model.

The asphaltene structure flows in a liquid-solid transition regime, reaching different

metastable states as energy, in stress form, is added to the system. When the energy is arrested

(when the shear rate is reduced, or the sample is at rest), a structural rearrangement is needed

to return to the initial energy state. It is likely that two types of structure have the same

average energy state when the fluid has been moderately stressed, so the architecture of the

structures would be a more reliable feature to analyze aging and irreversibility when higher

stress is applied.

.

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01 1,E+00 1,E+01

Modu

li [

N.m

]

ω [rad/s]

G'- 5 min

G''- 5 min

26

Conclusions

The non-Newtonian behavior of heavy crude oil was studied using hysteresis loop tests,

which have proven to be a simple and effective tool for observing thixotropy. The loop test

was performed at 25ºC on samples with different stress histories, and by comparing the

hysteresis area formed by breakdown and buildup stages, it was concluded that thixotropy

increases after flow and rest stages. The 30-day loop showed an irreversible change in

viscosity, which means that the effects of aging of the structures became visible for highly

stressed samples.

Flow tests were performed at temperatures between -5 and 30ºC to determine the contribution

of waxes to the thixotropy observed in the crude oil. The absence of a gel point discards the

waxes as the cause of thixotropy. Loop tests in which the crude oil composition was modified

by evaporation of maltenes show a direct relation between thixotropic area and asphaltenes

content, reinforcing the hypothesis that the self-assembly nature of the asphaltenes may be

responsible for the structures that show reversible and shear rate dependent changes.

Oscillatory tests were introduced to study the viscoelasticity of stressed samples. The moduli

do not change after a prolonged deformation and 12 hours of rest. The system manages to

return to its equilibrium condition after stress induced by the peak-hold, showing evidence

of reversible structural change that may be responsible for buildup and viscosity recovery in

flow tests. Viscoelasticity was also studied using creep tests. In the first stage, the crude oil

is deformed like a Burgers viscoelastic liquid. However, in later stages, with different

imposed stress and recovery times, the crude oil shows instantaneous solid-type deformations

in early times (1 second). These deformations correspond to the metastable states that the

structure reaches when it is reorganized and can be studied from percolation theory

perspective, which explains abrupt transitions between states when the combination of nodes

and bridges (asphaltenes and bonds) is close to a critical probability.

The study of two-dimensional structures provides information to facilitate the construction

of three-dimensional models such as flocculated networks formed by asphaltenes. Interfacial

27

rheology tests were performed in a dodecane-water system at a surface coverage of

1,5 𝑚𝑔 𝑚2⁄ to analyze the interactions in two dimensions. The structure has a linear

viscoelastic zone for strain around 10% and frequencies between 0.01 and 1 rad/s. The

moduli do not change between flow tests that go up to 10 rad/s, probing reversibility. The

interface viscosity decreases three orders of magnitude and stabilizes at a shear rate of 2 𝑠−1.

It is likely to be the point at which the balance between fragmentation and aggregation is

reached. When the structure flows close to its solid-liquid transition, as expected for the

concentration of 1,5 𝑚𝑔 𝑚2⁄ , it shows a soft glassy behavior in which the structure reaches

multiple metastable states as it is energy added to the interface, when this energy is retired

the interface tends to return to its state of lower energy. Given the absence of hysteresis in

the interface flow tests, it is possible to say that the buildup time of the structure is similar to

the breakdown time.

Fig. 13 Possible connections between the percolation theory and the phenomena observed in the five types of tests performed

The viscosity and the measurements of the moduli are an indicator of the average energy

state of the system. The same energetic state can be reached by different structures, which is

why it is important to study the evolution of the architecture of these structures over time.

The definition of the architecture of the asphaltene aggregates would allow predicting

thixotropy, viscoelasticity and aging phenomena observed in this work without the numerous

input parameters required by existing models.

28

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