Prior assessment of CO2 leak rate through cracks sealed by nanoparticle gels

Abstract

The leakage of hydrocarbon fluids through cracks in the annular cement and CO 2 storage is a major concern to the Petroleum Industry. A significant risk is posed when repairing leakage in a micro annuli channel with smaller apertures. A low-viscosity sealant that can generate a long-lasting resilient seal is desired. The solution to sealing these channels might lie in a novel application using nano-silica Gel. In this study, laboratory tests were carried out to examine the capabilities of nano-silica gels to seal the cracks. Analyzing its rheological property, the gel strengths of nano-silica gels were found to increase with an increase in nano-silica concentration. Additionally, it was discovered that as the concentration of nano-silica increases, the sealing and leakage pressures, defined as the pressures before and after water breakthrough, respectively, increase as well. With a typical 15% concentration of nano silica in gel, a sealing pressure gradient of 80.2 psi/in and a leakage pressure gradient of 30 psi/in at a leaking rate of 1 cc/min were noted. To validate the validity of the experimental results, a mathematical model was developed to predict the leakage rate of sealed fractures. The model suggests that the young’s modulus of sealant is a key property of nano-sealants and further investigations are needed to validate the mathematical model for quantitative use. This study suggests a novel strategy for enhancing cement zonal isolation and reducing cement failure in oil and gas sector.

Full text

Vol.:(0123456789)
1 3
Journal of Petroleum Exploration and Production Technology
https://doi.org/10.1007/s13202-023-01626-1
ORIGINAL PAPER-PRODUCTION ENGINEERING
Prior assessment of CO2 leak rate throughcracks sealed
bynanoparticle gels
OlatunjiOlayiwola1 · VuNguyen1· NingLiu1· BoyunGuo1
Received: 15 September 2022 / Accepted: 13 March 2023
© The Author(s) 2023
Abstract
The leakage of hydrocarbon fluids through cracks in the annular cement and CO2 storage is a major concern to the Petroleum
Industry. A significant risk is posed when repairing leakage in a micro annuli channel with smaller apertures. A low-viscosity
sealant that can generate a long-lasting resilient seal is desired. The solution to sealing these channels might lie in a novel
application using nano-silica Gel. In this study, laboratory tests were carried out to examine the capabilities of nano-silica
gels to seal the cracks. Analyzing its rheological property, the gel strengths of nano-silica gels were found to increase with an
increase in nano-silica concentration. Additionally, it was discovered that as the concentration of nano-silica increases, the
sealing and leakage pressures, defined as the pressures before and after water breakthrough, respectively, increase as well.
With a typical 15% concentration of nano silica in gel, a sealing pressure gradient of 80.2 psi/in and a leakage pressure gradi-
ent of 30 psi/in at a leaking rate of 1cc/min were noted. To validate the validity of the experimental results, a mathematical
model was developed to predict the leakage rate of sealed fractures. The model suggests that the young’s modulus of sealant is
a key property of nano-sealants and further investigations are needed to validate the mathematical model for quantitative use.
This study suggests a novel strategy for enhancing cement zonal isolation and reducing cement failure in oil and gas sector.
Keywords CO2-leak· Nano-particle gel· Sealants· Test· Modeling
List of symbols
Latin letters
B Hydraulic Aperture, in
Cf Fracture Conductivity, mD-ft
D Core Diameter, in
E Young’s Modulus, psi
h Fracture height, in
L Fracture length, ft
Dp Pressure Drop, psi
dp/dx Pressure gradient, psi/ft
p(x) Fluid pressure at point x, psi
p1 Flow inlet pressure, psi
p2 Flow outlet pressure, psi
Q Fluid Flowrate, ft3/s
v(x) Fluid Velocity, ft/s
wc Channel width, in
wf Fracture width, in
Greek letters
µ Dynamic viscosity, cp
e Strain, psi
s Stress, psi
API American petroleum institute
CCS Carbon capture and storage
WCR Water cement ratio
XRF X-ray fluorescence
Abbreviations
API American petroleum institute
CCS Carbon capture and storage
WCR Water cement ratio
XRF X-ray fluorescence
Introduction
One of the most crucial activities in drilling operations is
cementing engineering. There are many variables at play
when a well fails. Because of changes in the downhole envi-
ronment, cement is vulnerable to cracking during a well's
life. It is never acceptable to compromise the integrity of the
cement during drilling or production operations. Leaks may
* Olatunji Olayiwola
olatunji.olayiwola1@louisiana.edu
1 University ofLouisiana atLafayette, Lafayette, USA

Journal of Petroleum Exploration and Production Technology
1 3
develop at any moment during an active life of the well if the
cement is not properly completed and abandoned. Through
channels within the cement or between the cement and its
surroundings, fluids (water or hydrocarbons) can migrate.
When the wellbore integrity is compromised, these channels
form, enabling formation fluids to move from the formation
to the surface. Water, gases, and hydrocarbon compounds
that escape through cement pathways endanger people and
the environment.
Carbon Capture and Storage (CCS) has become a key
technology used for cutting CO2 emission to the atmos-
phere to mitigate climate change. However, the leakage of
injected CO2 is one of the major environmental concerns.
CO2 leakage through wells is one of the critical risks identi-
fied in CO2 sequestration operations (Duguid etal. 2013;
Zhang and Guo 2014; Duguid etal. 2017; Guo 2017). The
leak is usually attributed to the “flow behind pipe” which is
referred to as fluid flow through channels/fractures/cracks
in the annular space between well casing and formation
rock. The channels are usually caused by insufficient place-
ment of cement in the annulus. The fractures/cracks can be
induced by the cyclic pressure and thermal loading in well
operations (Goodwin and Cook 1992; Nygaard etal. 2014;
Weideman and Nygaard 2014). Goodwin and Crook (1992)
analyzed cement sheath stress failure in various conditions.
CO2 attack to the cement sheath can accelerate the develop-
ment of the cracks (Nguyen etal. 2021; Zhang etal. 2021).
The flow channels in the wellbore annulus can be plugged
using the conventional cement squeezing technique because
of the large cross-sectional areas open for cement slurry to
flow (Benge 2009; Hossain and Amro 2010). Guo etal.
(2017a, b) predicted radial cracks in cement sheath due to
casing internal pressure. Duguid etal. (2017) found radial
cracks of approximately 1/2-mm-wide and 35- mm-deep
in the sidewall-cores from an actual cement sheath. These
cracks are too narrow to be sealed by cement squeezing
because of the high pressure required to inject cement slurry
of high viscosity. Recently nano-particle solutions have been
studied to seal the cracks. The leak-off assessment of nano-
particle-stabilized CO2 foams for fracture/crack applications
was performed by Nguyen etal. (2022), Fu etal. (2022),
and Olayiwola etal.(2022a). Olayiwola etal. (2022b) dem-
onstrated promising properties of nanoparticle-based gels
to seal fractures in lab conditions. However, gels are soft,
and their sealing performance depends on their mechanical
properties such as young’s modulus. It is highly desirable to
know the effect of gel mechanical properties on its sealing
performance under different crack conditions.
Investigation of CO2 leak rate through fractures/cracks
sealed by nanoparticle falls in the study area of leak of seal
with a gap. The fluid flow through the gap is fundamentally
described by the Hagen–Poiseuille equation (Jousten 2008).
Lorenz and Persson (2009) presented experimental result for
the leak rate of a rubber seal with rectangular cross-section.
They compared the result to a theoretical model developed
based on the percolation theory and their contact mechan-
ics theory. They found good agreement between theoretical
model and experiment. Persson (2022) investigated fluid
leakage around a rubber seal with cylindrical cross-section
based on the percolation at the rough contact area. Uniform
fluid pressure distribution along the gap was assumed in
these investigations. This assumption is not valid for study-
ing the fluid leak rate through fractures/cracks sealed by
nanoparticle because the fluid pressure varies greatly along
the fractures/cracks of hundreds of meters long in the axial
orientation of wellbore.
An experimental set-up was used to look at the ability
of Nano-silica gel to stop fluid leakage through channels
in the cement. There hasn't been much research done on
crosslinked micro-use gels in cement zonal isolation. Addi-
tionally, this study will provide petroleum engineers with
a fundamental understanding of how to apply Nano-silica
sealer for wellbore integrity applications. In this study,
it will be examined how the concentration of nano-silica
affects the gelation characteristics of the sealant, its ability
to block routes, and its capacity to stop leakages. The labora-
tory investigations include rheological testing, gelation time
predictions, blocking effectiveness, and mechanical proper-
ties. A prior assessment of CO2 leak rate through fractures
sealed by nano-particle gels was carried out in this study
based on experimental investigation and mathematical mod-
eling. It was found that the CO2 leak rate is very sensitive to
fracture/crack width and young’s modulus of gel, influenced
by fracture/crack height and CO2 viscosity, and controlled
by pressure differential.
Methodology
Materials
Nano‑silica gel
Nouryon Pulp and Performance Chemicals Inc. supplied the
nano silica used in this investigation. DI water and sodium
chloride from Fisher Scientific were also used in this experi-
ment. The nano silica utilized has grains that are 50nm in
size. According to the supply specification, the silica that is
utilized is a colloidal, aqueous dispersion that is alkaline and
has a solids content of around 50%. The surface-modified
amorphous silica particles have a negative surface charge.
The silica particles have a broad particle size variety, are
discrete, and have a smooth, spherical form. For optimal
performance as an emulsion stabilizer, the particles' surfaces
have been treated with silanes. The dispersion physically
resembles a white liquid that is a little bit more viscous than

Journal of Petroleum Exploration and Production Technology
1 3
water. The material was used exactly as it was supplied from
the company, with no chemical changes.
Brine
A brine solution was prepared using commercially available
NaCl with a purity of 99.99 percent.
Cement core
All the cement core specimens used in this investigation
were prepared with Class H cement and distilled water.
Using a gas pycnometer, the specific gravity of the cement
was determined to be 3.18. X-ray fluorescence spectroscopy
was used to determine the chemical composition of Class-H
cement (XRF).
Preparation ofcement paste
The cement slurry was blended at room temperature using a
bottom-drive blender with two speeds. A precise amount of
water was poured to the blender, and dry cement was added
in a consistent manner. The blender was then combined on
low speed for around 15s. The blender was then covered,
and the mixing continued for an additional 35s at high speed
(API RP 10B-2). According to API specification 10A for
API cement Class-H, the cement slurry's water/cement ratio
(WCR) was 0.38. (API 2010).
Fracturing thecement cores
A two-inch diameter tube was filled with the cement slurry
inside, closed on one end, and left open on the other for 72h
to solidify, resulting in the split cement cores. The cement
core was taken out of the pipe after it had hardened. The dry
core was then gently pressed against the tile-saw blade while
it was running to split it into two halves. Using proppant or
thin-metal plate, the two pieces of the core are then sepa-
rated by the necessary fracture width, as depicted in Fig.1.
Gel preparation
A simple experimental procedure was used to create silica
gels at room temperature using Nano silica concentrations
ranging from 15 to 40%, salt concentration, and water, as
illustrated in Fig.2. The required concentration of nano-sil-
ica particles was initially synthesized. The 3%-concentrated
brines were then added to the nano-silica particle and given
time to gel. The rheological and elastic characteristics of the
resulting gel particles were then assessed using an Anton
Par Rheometer. Figure2a and b shows the appearance of
the nano-silica gel before and after gelation. An OFITE 900
Rheometer was used to examine how the concentration of
nano-silica affects gel's rheological and sealing properties.
Nano silica particles are dispersed in a stable inorganic
aqueous solution as colloidal silica. It is distinct from other
organic solutions due to the substance's ease of handling
and environmental friendliness. The solution exists in ambi-
ent conditions as a low viscosity fluid, like water, which is
advantageous for simple pumping. Colloidal silica is thought
to gel because of particle collision, bonding, and aggrega-
tion into long-chain networks. pH, salts or cations, particle
concentration level and temperature can cause Nano silica
solution to gel. Hydroxyl ion (OH−) is drawn to the surface
of silica in aqueous solution and forms the silanol (Si–OH)
group.
The salt solution injected during the gelation process
functions as an activator or accelerator. The generated SiO
Fig. 1 A cut cement core sec-
tion. a Fractured core before
gel placement, b fractured core
filled with gel
Fig. 2 Appearance of nano silica gel. a Before Gelation b After Gela-
tion

Journal of Petroleum Exploration and Production Technology
1 3
group begins connecting with Si–OH to produce the siloxane
bond when an activator is present. Finally, it is possible to
imagine that long chain siloxane linkages arise while silica
gelation takes place. Gelation happens when silica particle
aggregation leads to the formation of a consistent 3D struc-
tural network. When fracture is induced in the well cement,
it may generate either macro or micro sized fracture depend-
ing on the degree of fracturing effect. Regardless of the frac-
ture size, nano-silica dispersion can penetrate through the
size of fracture because of its initial low viscosity.
Injectivity test
The gel's passage during the cement setting is described in
Fig.3. The apparatus consists of a 2-inch Core holder, two
syringe pumps, and two accumulators with pistons. A 0.05-
inch proppant was inserted between the two pieces of the
broken core to gauge the fracture's width. Then, to create
a complete core, the two sections are taped together. Then,
confining pressure is applied to the core holder's rubber
sleeve to stop fluid leaking around the fractured core and
to simulate the confining pressure circumstances. Brine and
nano-silica gel were injected into the core-holder through the
accumulator. Pressure transducers were installed at the cell's
intake, middle, and outflow to track the gel's performance
along the fracture.
The Nano silica solution and the brine were co-
injected in the laboratory core flooding test until the gel
was formed inside the fractured core. An injection test
was carried out after the cement core's cracks had been
sealed with nano silica gel. The gel strength created by
Nano silica imbedded in cement cracks during injection
prevented fluid migration through the cracks until the gel
lost its sealing strength, at which point water started to
leak through. The pressure that Nano silica gel develops
to close the fractures is referred to as the sealing pressure,
whereas the pressure that develops after breakthrough is
referred to as the leakage pressure.
The duration of the fluid blockage by Nano silica gel is
a function of the gel strength developed during gelation
process. This strongly support the fact that colloidal sil-
ica gel has enough strength to resist holdback pressure of
reservoir fluid to escape through the fracture. The funda-
mental benefit of nano silica in maintaining well integrity
is that the gelation mechanism accelerates in downhole
conditions. The production of gel may be sped up when the
solution enters the fracture aperture and meets the cement
surface due to the possibility of bivalent Ca2+ leaching out
of the cement. To simulate the initial condition where the
cement concrete is already filled with formation fluid (nor-
mally water) before the silica injection, brine was pumped
into the core-holder using an accumulator. After the brine
was in place, a steady flow rate of 1cc/min of nano-silica
particle gel was injected into the core-holder.
The gel injection was continued till the formed gel
was seen at the exit and the injection pressure differen-
tial across the fracture was steady. The injection pressure
was measured by pressure sensors, and the angle of gel
propagation across the fracture was monitored with a high-
resolution camera. To test the gel's water sealing perfor-
mance, brine was introduced into the gel packed fracture
after the gel was set in place. The brine was injected at a
continuous flow rate of 1cc/min through an accumulator.
The brine injection was continued till the pressure of the
brine injection was stabilized. Pressure transducers were
used to record pressure data in the testing period as shown
in Fig.3.
Equation(1) is applied to calculate the fracture conduc-
tivity during the injection test.
Fig. 3 Experimental set-up for the injectivity test

Journal of Petroleum Exploration and Production Technology
1 3
where Q (flowrate) in cc/min, µ (viscosity) in cp,
L(corelength)
in inch, D (core diameter) in inch and
Cf
(frac-
ture conductivity) in md-ft and, DP (pressure differential) in
atmospheric pressure units (psi).
Equation(2) is used to determine the hydraulic aperture
of the fracture, B.
In this equation, Hydraulic Aperture, B is expressed in
inches, Flowrate, Q in cubic centimeters per minute, Vis-
cosity, µ is expressed in centipoise (cp), Length of the core,
L is expressed in inches, fracture width, wf is expressed in
inches, and the pressure drop, DP is expressed in atmos-
pheric pressure units (psi).
Leakage rate model
Figure4 depicts a process of fluid-channeling in a gel-filled
fracture. Due to the deficiency of gel placement in the frac-
ture, fluid can enter the fracture under fluid pressure. Fluid
squeezed into the gel-sealed fracture compresses the gel and
induces stress and strain in the gel. The shrinkage of gel in
the direction perpendicular to the fracture is equal to the
width of fluid channel:
where wc(x) is the channel width at point x, e(x) is the
strain in the gel at point x, and wf is fracture width which is
assumed to be constant. The strain is expressed as
(1)
C
f=
8.036Q
𝜇
L
DΔp
(2)
B
=0.3937
(
0.00242Q𝜇L
w
f
ΔP
)1∕3
(3)
wc(x)=
𝜀
(x)wf
where s(x) is the stress at point x, p(x) is the fluid pressure at
point x, and E is Young’s modulus of gel with nanoparticles.
Substituting Eq.(4) into Eq.(3) gives:
Hagen–Poiseuille equation (Jousten 2008) takes the fol-
lowing differential form in consistent units:
where the fluid velocity can be expressed as
where h is the height (or depth) of fracture which is assumed
to be constant. Substituting Eq.(7) into Eq.(6) yields:
Substituting Eq.(5) into Eq.(8) gives:
which is integrated as
(4)
𝜀
(x)=𝜎
(x)
E
=
p(x)
E
(5)
w
c(x)=
w
f
E
p(x)
.
(6)
dp(x)
dx =−
12
𝜇
v(x)
w2
c
(x)
(7)
v
(x)=
Q
hw
c
(x
)
(8)
dp(x)
dx =−
12
𝜇
Q
hw3
c
(x)
.
(9)
dp
(x)
dx =−12𝜇E
3
Q
hw3
f
p3(x
)
Fig. 4 Fluid-channeling in a
gel-filled fracture. a Fracture
before gel placement, b fracture
filled with gel, c fluid squeezed
into gel-sealed fracture, d
pressure-induced stress in gel

Journal of Petroleum Exploration and Production Technology
1 3
where p1 is the pressure at flow inlet point (x = 0), p2 is the
pressure at flow outlet point (x = L), and L is fracture length.
The result of the integration is
This equation takes the following form in U.S. oilfield
units.
where pressures p1 and p2 are in psi, dynamic viscosity m
is in cp, young’s modulus E is in psi, flow rate Q is in ft3/s,
fracture length L is in ft, fracture height (or depth) h is in
inch, and fracture width wf is in inch. The flow rate is solved
from Eq.(12) to obtain:
Results anddiscussions
This section summarizes experimental and model results of
tests for gel strengths, sealing pressures, and leakage pres-
sures of gels with different concentrations of nano-silica.
The gel's capacity to plug cement fractures and reduce fluid
migration through fractured cement was investigated using
the Gel-Cement Plugging Efficiency setup. Using Eqs.(1)
and (2), the fracture conductivity and the hydraulic aperture
could be estimated.
The gel injection pressure is shown in Table1 for vari-
ous flowrates. The cement fracture that was employed in
this study has a diameter of 2 inches, a length of 10 inches,
and a width of 0.05 inches. The pressure of the gel injection
(10)
p
2
∫
p1
p3(x)dp(x)=−
12𝜇E3Q
hw3
f
L
∫
0
dx
(11)
p
1=4
√
p4
2+
48𝜇E3QL
hw3
f
.
(12)
p
1=4
√
p4
2+
0.144𝜇E3QL
hw3
f
(13)
Q
=
6.94hw
3
f
(
p
4
1
−p
4
2
)
𝜇E
3
L
.
increases with flowrate. The greatest flowrate of 4cc/min
was used to reach the peak pressure of 265 psi. When the
flowrate was reduced to 3, 2, and 1cc/min, the injection
steady pressure fell to 173, 113, and 75 psi, respectively. The
Darcy law of flow, which states that pressure drop increases
as flowrate increases, is supported by these data. Average
hydraulic aperture size and fracture conductivity for the
cement core were found to be 0.64 md-ft and 0.08 inches,
respectively, according to the results provided in Table1.
Gel strength
Figure5 demonstrates that the growth of gel strength is
accelerated by increasing the concentration of nano-silica.
The observed gel solution produced a gel strength of 504
100lb/ft2 for a 15% nano-silica concentration, and when the
nano-silica content was increased to 40%, the gel strength
increased to 8,492.6lb./100ft2. A saturation effect may be
seen in the leveling-off above the nano-silica concentration
of 35%. Due to the increased molecular weight brought on
by the nano-silica particles' presence in the fluid, the gel's
texture thickens as their concentration rises. The hydroxyl
groups included in nano-silica particles' structure make
it easier to create a filler network in polymer chains and
increase gel strength. (Zareie etal. 2019). This is owing to
the Nano-silica particle's aggressive filler network, whose
rheological property increases as its concentration in the
solution increases (Havet etal. 2003). This increase in
rheological feature raises the corresponding gel strength by
increasing the ionic strength of the mixture solution (Gal-
lagher and Mitchell 2002).
Sealing andleakage pressures
Figure6 displays the pressure curve for a fracture that was
sealed using a 15% concentration nano-silica solution dur-
ing the post-gel brine injection test. The sealing pressure,
or greatest point of injection pressure, was 810 psi prior to
Table 1 Experimental result at different flowrates
Inj. Rate (cc/min) Stabilized pres-
sure (psi)
Cf (md − ft) B (inch)
1 75 0.54 0.073
2 113 0.71 0.081
3 173 0.69 0.08
4 265 0.61 0.076
Average 0.64 0.08 Fig. 5 The gel strength versus nano-silica concentration

Journal of Petroleum Exploration and Production Technology
1 3
brine breaching the sealant. The strength of the sealant is
determined by this sealing pressure, which is equivalent to
802/10 = 80.2 psi/inch. Following the brine breakout, the
injection pressure dropped to a steady leakage pressure of
300 psi. The flow must be kept at a leakage pressure gradi-
ent of 300/10 = 30 psi/inch. Therefore, the sealed fracture
leaks at a rate of 1cc/min when there is a 300-psi pressure
difference.
The sealing pressures of gels with various amounts of
nano-silica are shown in Fig.7. It demonstrates how the
sealing pressure rises as the concentration of nano-silica
does. Under the 25% nano-silica concentration, the leveling-
off indicating the saturation effect is not visible. Figure8
depicts the leakage pressures of gels containing different
concentrations of nano silica. It shows that the leakage pres-
sure increases as nano silica concentration does. There is no
leveling-off at a nano silica concentration of 25%, indicating
the saturation effect.
Predicting the leakage rate with Eq.(13) requires the data
for young’s modulus of sealant, the nano silica gel in this
case. Unfortunately, this piece of information is not found
from literature. Figure9 presents a sensitivity test of fracture
leakage rate against the uncertain value of Young’s modulus
of sealant. It indicates that the leakage rate declines sharply
with Young’s modulus according to Eq.(13). If the math-
ematical model is correct, the Young’s modulus of the nano-
silica gel should be about 3050 psi for sealant with 15%
nano-silica gel presented in Fig.6.
Conclusions
A series of tests were carried out in this study to investi-
gate the sealing properties of nano-silica gels for sealing
underground fractures/cracks near oil/gas production and
CO2 sequestration wellbores. The following conclusions
are drawn.
1. A resilient Silica seal can be created at room temperature
by mixing Nano Silica solutions, Salt, and Water.
2. Up to a concentration of 35% nano silica, the gel strength
of nano silica gels gets stronger with increasing nano
silica concentration. Above the 35% concentration, the
line flattens out, illustrating the saturation effect.
3. The concentration of nano-silica causes an increase in
the sealing pressure, which is the maximum pressure
before water breakthrough. Under the 25% nano-silica
concentration, the leveling-off reflecting the saturation
effect was not seen. The sealing pressure gradient is 80.2
psi/in when nano-silica is present in gel at a typical con-
centration of 15%.
4. As the concentration of nano-silica increases, the leak-
age pressure—defined as the pressure that stabilizes
after water breakthrough—increases. Under a 25% nano-
silica concentration, the trend remains unchanged. To
produce a leakage rate of 1cc/min with a standard 15%
concentration of nano-silica in gel, a leakage pressure
gradient of 30 psi/in is needed.
Fig. 6 Pressure profile recorded during a brine injection test
Fig. 7 The maximum sealing pressures for nano-silica gels of differ-
ent concentrations
Fig. 8 The leakage pressures for nano-silica gels of different concen-
trations

Journal of Petroleum Exploration and Production Technology
1 3
5. To anticipate the leakage rate of sealed fractures, a math-
ematical model was created. According to the model, the
leakage rate of a sealed fracture is directly proportional
to fracture height, fracture width raised to the third
power, pressure differential raised to the fourth power,
and inversely proportional to fluid viscosity, Young's
modulus of sealant raised to the third power, and fracture
length.
6. The mathematical model's sensitivity analysis shows that
as the sealant's Young's modulus is raised, the leakage
rate of sealed fractures rapidly decreases. To efficiently
seal cracks, strong sealants with high Young's modulus
values need be created.
7. Further investigations are needed to measure the
Young’s modulus of nano-particle gels and validate the
mathematical model for quantitative use. Without fur-
ther validation, the model may only be used qualitatively
for analyzing factors affecting the sealing performance
of fracture sealants.
Acknowledgements The authors are grateful to BIRD for supporting
the project “Safe, sustainable, and resilient development of offshore
reservoirs and natural gas upgrading through innovative science and
technology: Gulf of Mexico – Mediterranean,” through Grant No.
EC-19 Fossil Energy.
Funding No external funding was used for this research work.
Declarations
Conflict of interest The authors declare that they have no known com-
peting financial interests or personal relationships that could have ap-
peared to influence the work reported in this paper.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
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