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 13ΒΊAPI 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 2 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.
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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
13ΒΊAPI 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 ππ
π2 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 10-22.3ΒΊAPI 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. ππ β« 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. ππ βͺ 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 nature10β12. 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 stages13β16. 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 (18Β°πΆ). 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 15.5Β°C ASTM D7042-19 18 954
Density (Β°API) at 15.5Β°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, β₯ 99%), for asphaltene precipitation, and N-dodecane (anhydrous,
β₯ 99%), used as oil phase in interfacial experiments, were purchased from Sigma-Aldrich
Co. Chloroform (HPLC grade, β₯ 99%), used for all asphaltene solutions. Ultra-pure water
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 25β 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 1π β1 to 3000π β1 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 28β31.
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 ΒΊC 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 ΒΊC was
performed. The procedure was repeated with an evaporation time of 3 days, but on the third
day, the flow sweep was performed until 100π β1 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 ΒΊC 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. MaltenesΒ΄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 25β in between evaporation rest periods of 1
day and 3 days. Initial and 1 day flow sweeps were performed from 1π β1 to 3500π β1 and back. 3
day flow sweep was performed from 1π β1 to 100π β1and 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 3000π β1.
So far, loop tests have shown a drop in viscosity at around 100π β1 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 rates36β39.
According to Fig. 2 at 3000π β1 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 oil13,14,40β42. 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 ΒΊC. However, different studies have shown that in the presence of asphaltenes,
12
gelation temperature and yield stress decreaseβbecause asphaltenes act as "weak points" in
paraffin gel-like structures43β47.
If thixotropy were the result of waxy gelled structures, there should be a gelation temperature
close to the testing temperature used until now (25 ΒΊC), so that the flow tests were affected
by the onset of paraffin gelation process.
Fig. 4 Temperature ramp from 30 ΒΊC to -5 ΒΊC. The shear rate was set at 1π β1 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 1π β1. The sample was first heated to 50 ΒΊC 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 [ΒΊC]
13
Fig. 5 Frequency sweeps with a 100% strain of heavy crude oil measured at 25β. 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 1000π β1, 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 πΊβ
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]
Gβ Initial 1.574 -1.540 10156.83
Gββ Initial 0.997 0.773
Gβ Post peak-hold 1.977 -1.781 423.03
Gββ Post peak-hold 1.001 0.783
Gβ 12 h 1.990 -1.447 387.95
Gββ 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.
πΊβ = βπΊβ²2 + πΊβ²β²2 πΈπ. (1)
πΊβ² =πΊ(ππ)2
1 + (ππ)2 πΈπ. (2) πΊβ²β² =
ππ
1 + (ππ)2 πΈπ. (3)
π =π
πΊ πΈπ. (4) π =
1
π πΈπ. (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 (100π β1) 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 ΒΊC.
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 Γ 10β4 ππ to 1 ππ 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 ΒΊC 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
πΎ(π‘)
π=
1
πΊ1+
1
πΊ2(1 β π
βπ‘π2β ) +
1
πΊ3(1 β π
βπ‘π3β ) + (β¦ ) β¦ +
π‘
π πΈπ. (6)
ππΎ
ππ‘= π [
πβπ‘
π2β
πΊ2π2+
πβπ‘
π3β
πΊ2π3+ (β¦ ) β¦ +
1
π ] πΈπ. (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 (Ο) 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 βnodesβ connected by βbridgesβ (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 others58β60.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
viscoelasticity65β67.
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 (Ξ») 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 Ξ» 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 ππ
π2, 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 Γ 10β5ππ, 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 Γ 10β1 π , 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 ΒΊC.
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
, Ξ΄
[ΒΊ]
Moduli
[N
.m]
Ο [rad/s]
b
G' G''
Tan (Ξ΄) Ξ΄
24
Fig. 11 Flow sweeps of asphaltenes from 1 Γ 10β3π β1 to 10 π β1 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 π β1. 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 π β1. 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
]
Ο [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 25ΒΊC 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 30ΒΊC 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 ππ π2β 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 π β1.
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 ππ π2β , 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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