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2024 SPE Journal 1
Experimental and Theoretical Studies of
Nano- Silica Solutions in Narrow Flow Paths
Applied to Sealing Cement Cracks
Md NahinMahmood1* , Vu VNguyen1 , and BoyunGuo1
1Department of Petroleum Engineering, University of Louisiana at Lafayette
Summary
A cement crack is a typical cause of oil and gas well failure. Cracks weaken cement, reducing zonal isolation and uid leakage.
Nanoparticle (NP) gels are being tested for fracture treatment. When crushed into cracks, the ow behavior of NP problem solutions
should be predicted. The potential ecacy of utilizing NP gels as a remedial measure for fractures is currently under investigation. It
would be advantageous to determine if the ow behavior of solutions for NP problems can be anticipated when they are compressed into
crevices. This study aimed to analyze the behavior of nano- silica solutions as they ow through ducts with rectangular cross- sections and
varying crack dimensions. The introduction of NP solutions into the core leads to a decrease in pressure, which suggests that the nano-
silica has been eectively transported through the crack. As the size of the fracture decreases, there is a corresponding increase in pressure
drops, while the ow rate experiences a concurrent increase. This study presents responses of a pressure gradient to uid concentration for
a range of fracture widths, heights, and ow rates. The prediction of laminar ow in ducts is based on the linear correlation between the
ow rate and the pressure gradient. Furthermore, the reduced pressure gradient indicates enhanced uid ow within the fracture because
of the amplied slot width. The uid ow model proposed by Guo et al. (2022) was utilized to conduct a comparative analysis with the
experimental data. Compared with test data, the model diers by roughly 90%. The technical cause of the ow model- observed data
discrepancies is unknown. The ow model did not account for friction between NPs- NPs and NPs- walls in rough ducts. An empirical
correlation has been found that quanties the ratio as a function of nonsilica solution ow rate, cross- sectional geometry parameters,
and nano- silica concentration. The correlation was calculated using nonlinear regression. The empirical relation and actual ratio have a
signicant correlation, as shown by R2 = 0.8965. In practice, Guo et al.’s (2022) hydraulic model’s pressure drops should be multiplied
by the empirical correlation’s ratio to reduce errors.
Introduction
Recently, solutions containing NPs have been identied as a feasible alternative within the oil and gas sector. The transportation of NPs
has been subjected to experimentation in diverse scenarios, such as enhancing the quality of drilling and completion uids, as well as
regulating conformance in enhanced oil recovery procedures. In the context of sealing naturally occurring ssures and cracks in building
materials such as oilwell cement concrete, inquiries arise regarding the movement of NPs within said gaps. The precise positioning of NPs
denotes their inuence on the eectiveness of sealing mechanisms. NPs are driven into crevices by diverse physical forces, such as grav-
itational, capillary, and viscous forces.
Various researchers have investigated the impact of NPs on the ecacy of drilling and completion uids, as well as on enhanced oil
recovery conformance operations. The transport of NPs in porous surfaces has been extensively investigated by several researchers,
namely, Tufenkji and Elimelech (2004), Li et al. (2008), Wang et al. (2012), Becker et al. (2015), Bianco et al. (2016), and Zhang et al.
(2016). The trajectory of spherical particles can be predicted through the usage of the Colloid Filtration Theory, and the mechanisms of
particle transportation and the Colloid Filtration Theory models’ usage for NPs in porous media are also investigated (Rajagopalan and
Tien 1976; Molnar et al. 2015).
Khalil et al. (2017) have recently published a scholarly article discussing the design and challenges associated with the applications of
NPs. Babakhani et al. (2017) conducted a comprehensive analysis of the continuum- based models and concepts pertaining to the transport
of NP in saturated porous media. The experimental ndings of Mohamadian et al. (2018) pertained to the rheological and ltration prop-
erties of drilling uids that were fortied with NPs and various additives. Yekeen et al. (2019) conducted a comprehensive literature
review on the potential uses of NPs in the hydraulic fracturing process of unconventional reservoirs. Gu et al. (2020) observed the mani-
festation of shear thickening eects of drag- reducing NPs in a reservoir characterized by limited permeability. Several continuum models
have been devised to correspond with diverse particle transport pathways, to align with empirical data. While 3D modeling tools may be
preferable, it is noteworthy that most continuum- based models have been constructed and validated using solely 1D column data
(Babakhani et al. 2017; Velimirovic et al. 2020). Recently, researchers have extensively studied the use of NPs to obstruct ow channels
and ssures in the cement sheath of a wellbore, as evidenced by several studies (Olayiwola et al. 2022a, 2022b; Guo et al. 2022; Nguyen
et al. 2022).
Cai et al. (2018) introduced a semianalytical framework that integrated the gravitational subsidence of spherical particles into their
hydraulic transport through ssures. Cai et al. (2021) used the model to replicate the motion and arrangement of particles in indigenous
fractures. The phenomenon of capillary force was disregarded due to the considerable spatial separation between the fractures, as noted
by Guo et al. (2022).
The electrostatic interactions of NPs with fracture surfaces, uid interfaces, and other particles have a signicant impact on their
movement (Babakhani et al. 2017). In contrast to conventional uids, the density contrast between the NPs and the host uid may result
in phase separation due to gravitational and inertial segregation, as reported by Lyon- Marion et al. (2017). The unique slip mechanism
*Corresponding author; email: nahin2582@gmail.com
Copyright © 2024 Society of Petroleum Engineers
Original SPE manuscript received for review 23 May 2024. Revised manuscript received for review 8 July 2024. Paper (SPE 223088) peer approved 17 July 2024.
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2024 SPE Journal2
exhibited by NPs results in distinct uid behavior compared with conventional uids, posing challenges in the modeling of NPs transport
through cracks.
Furthermore, investigations have been carried out on the usage of silica NPs in polymer gel. The usage of the technology has been
observed in various domains, including but not limited to reinforcing and remediating wellbores (Shamlooh et al. 2019). Adibnia and Hill
(2017) examined the eects of the integration of silica NPs into a polymer gel and observed that this process would result in an augmen-
tation of the gel’s cross- linking. The rheological properties of the polymer gel are improved as a result of the hydrogen interaction
between the silica NPs and the polymer (Amir et al. 2022). According to Liu et al. (2017), there was an observed augmentation in the
thermal stability and strength of the gel. The augmentation of gel strength can be attributed to the cross- linking stage, wherein NPs amass
and disseminate throughout the meshes of the gel structure and polymer chain clusters. The enhancement of thermal stability in gels can
be attributed to the integration of silica NPs into the gel solution, as well as the augmentation of hydroxyl groups (OH) on their surface.
The placement of NPs poses a challenge due to the ow resistance generated by constriction in the channels and fractures. The chal-
lenge of sealing cement fractures becomes notably arduous when the transfer of NP to the downhole is a signicant concern. Several
research groups, including Beryani et al. (2020), Dong et al. (2020), and Johnson et al. (2020), have conducted investigations on the
dispersion of NPs in porous media. The study conducted by Song et al. (2019) aimed to investigate the impact of NP size on their transport
in porous media. On the contrary, Ma et al. (2020) conducted a study to explore the inuence of NP shape on NP transport. During a
tumbling event, the translational and rotational pathways of rod- shaped NPs undergo a transformation in their morphology. Tang and
Cheng (2018) have concluded that the movement of NPs in a porous medium is impeded by the electrostatic attraction between the NPs
and the medium, acting as a barrier. This conclusion was reached following an investigation conducted by the authors.
The cement sheathing utilized for wells features channels and ssures that exhibit a rectangular cross- sectional conguration, akin to
slots. The cross sections in question were classied by Guo et al. (2022) as either rectangular or bow- shaped. The usage of bow- shaped
cross sections was limited to the computation of cross sections and the determination of average velocities. By utilizing a well- established
equation that relies on the mean uid velocity, we successfully computed the reduction in pressure. Spiga and Morino (1994) demon-
strated the existence of a symmetrical solution for the velocity prole in laminar ow through ducts with rectangular cross sections. The
2D velocity distribution can be characterized by means of the nite Fourier transform. The authors asserted that their methodology
yielded numerical values with greater precision compared with the velocity distributions commonly used in existing literature. Regrettably,
the intricacy of the methodology suggested by Spiga and Morino (1994) results in infrequent application within the realm of engineering
analysis. Guo et al. (2022) developed a model to estimate uid ow by analyzing the velocity distribution in 2D cracks with nite heights.
The Hagen- concept Poiseuille’s method for laminar ow is utilized to construct a 1D velocity prole for uid ow between two parallel
plates. It is anticipated that this approach, as outlined by Jousten (2016), will produce more precise results compared with the conventional
method.
Before the studies conducted by Guo et al. (2022) and Nguyen et al. (2022), limited research had been undertaken to forecast the
movement of NPs within ssures that are signicantly narrower than those generated by hydraulic and natural processes. Later, the work
of Nguyen et al. (2022) was extended by Mahmood et al. (2023) with more data involving a combination of dierent fracture dimensions.
The study conducted by Zhang et al. (2019) demonstrated that the presence of capillary pressure within cement fractures can result in
signicant uid imbibition.
In their experimental study, Nguyen et al. (2022) and Mahmood et al. (2023) investigated the ow behavior of nano- silica solutions in
rectangular ducts. According to the research ndings, an elevation in the volumetric ow rate results in a corresponding rise in the pres-
sure gradient, in accordance with the Hagen- Poiseuille correlation for the laminar ow of Newtonian uid within tubes. The observed
pressure decrease exhibits a notable deviation from the anticipated values as per the models. The usage of the hydraulic diameter concept
in conjunction with the conventional hydraulics equation (Guo et al. 2017) provides an accurate depiction of the experimental results.
Nevertheless, it is postulated that the observed disparities in the outcomes can be attributed to the omission of NPs- NPs and NPs- wall
frictions in the investigations conducted by Jousten (2016) and Guo et al. (2022). The ndings of this study suggest that the concentration
of nano- silica did not have a signicant impact on the observed pressure drop. The pressure drop exhibited minimal uctuations in
response to alterations in the concentration of nano- silica.
This study represents an extension of the work conducted by Mahmood et al. (2023), wherein varying fracture widths and heights (i.e.,
slot widths) were examined for dierent concentrations of nano- silica solution. According to the ndings of previous research by
Mahmood et al. (2023) and Nguyen et al. (2022), the concentration of nano- silica solution was observed to have a negligible impact on
the pressure drop of the system. The study used nano- silica solutions of 5%, 10%, 15%, 20%, 25%, and 30% concentrations. The research
involves the injection of a uid into a sandstone core featuring varying fracture widths (0.03, 0.06, and 0.09 in.) and fracture heights, also
known as slot widths (1, 0.5, 0.25, 0.125, and 0.09 in.), with the aim of examining the pressure drop responses across dierent fracture
dimensions. Ultimately, the experimental ndings were juxtaposed against the hydraulic model proposed by Guo et al. (2022). Comparable
ndings indicate that signicant disparities are observed when compared with the model proposed by Guo et al. (2022); however, these
disparities remain consistent, with an approximate rate of more than 90%, regardless of fracture dimensions. The ndings indicate that the
observed dierences of the experiments are signicant in magnitude and that is why their ratio is calculated. Sensitivity analyses of this
ratio with four observed parameters (fracture height, fracture width, ow rate, and concentration) are carried out to identify any possible
correlation among them. A nonlinear regression equation has been formulated by analyzing all the data and the plot between the correlated
and actual ratio depicts an R2 of nearly 90% which justies the credibility of the observed experimental data and the model used to com-
pare them.
The result of this work should be used with caution owing to its limitations. It is understood that the properties of NP solutions change
beyond the gelation time. It is strongly recommended that the gelation time be measured. Fluid properties, especially viscosity, should be
measured to ensure that they are within the range of this work.
Working Procedure
The initial step of the experimental procedure involves bisecting a cylindrical core with a length of 22 in. and a diameter of 2 in. To rep-
licate a crack, a pair of metal plates are used to generate a gap of varying widths (0.03, 0.06, and 0.09 in.) between the edges of the pre-
viously sectioned core. The width of the ow area corresponds to the fracture height, also known as the slot width. The subsequent phase
involves the formulation of nano- silica solutions with varying concentrations of 5%, 10%, 15%, 20%, 25%, and 30% w/v. The laboratory
has conducted measurements on the viscosity of the solution, which ranges from 1.1 to 2.1 cp. The core has been placed within the core
holder, and subsequently, the Nano- silica solution is introduced into the accumulator through a pumping mechanism. The system is now
prepared for execution. The injection of the nano- silica solution is carried out via the “crack” at specic ow rates (5, 7.5, 10, and
12.5 mL/min) after the application of 400 psi of conning pressure. During the waiting period for the pressure drop to reach a stable state,
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2024 SPE Journal 3
data are collected at 1- minute intervals while injecting the uid. The procedure of the investigation is depicted in a sequential manner in
Fig. 1.
Fig. 1—Experimental flow chart of the research.
Experimental Setup
Experimental Apparatus. Fig. 2 depicts the fundamental design of the experiments conducted in this investigation. The identical
experimental design has been used in two prior investigations carried out by Nguyen et al. (2022) and Mahmood et al. (2023). The core
holder is a crucial component of the experimental system, as it is responsible for holding a sandstone core sample that has undergone
cracking while being subjected to conning pressure. The experimental arrangement utilized in the laboratory is depicted in Fig. 3. The
experimental procedure involves the insertion of a fractured cement core into a core holder, which is subjected to a conning pressure of
400 psi sourced from a nitrogen tank. Upon storage of the nano- silica solution in the accumulator, the Isco Pump will exert pressure to
propel it through the fractured cement core. Meanwhile, the computer system will record and monitor the pressure drop at predetermined
intervals via the pressure transducer.
Core Preparation. The cylindrical core, measuring 22 in. in length and 2 in. in diameter, was longitudinally divided and bisected. The
process of partitioning and reinforcing the two segments necessitated the usage of metallic material, specically lead sheet, which was
inserted along the peripheries as depicted in Fig. 4. The experiment involved the usage of metal plates with varying thicknesses of 0.03,
0.06, and 0.09 in., which were utilized as the fracture width or w. These plates were placed on each edge to generate a slot width or fracture
height that ranged from 0.09 to 1 in. Subsequently, the two parts of the core were axed together using adhesive tape, as depicted in
Fig. 5. The nonsupported area of the pieces forms a gap, resembling a crack.
Preparation of Nano-Silica Solution. The preparation of nano- silica with varying weight- per- volume (w/v) concentrations involved
the dilution of the initial nano- silica solution (w/v = 50%), as illustrated in Fig. 6. The viscosity of the solutions was measured in the
laboratory, resulting in values ranging from 1.1 to 2.1 cp. The nano- silica particle size used in this study is 50 nm.
Results
The experiment involved conducting injectivity tests utilizing nano- silica solution of varying concentrations and volumetric ow rates.
The action has been undertaken with the purpose of investigating the factors that contribute to inducing a decrease in pressure and gaining
further insights into the dynamics of uid movement within cement fractures. The experiments were conducted under a conning pressure
of 400 psi. The study investigated various crack dimensions, specically fracture width (w) of 0.03, 0.06, and 0.09 in., and fracture height
(h) of 1, 0.5, 0.25, 0.125, and 0.09 in. The volumetric ow rates of 5, 7.5, 10, and 12.5 mL/min were utilized in the investigation. The
height of a fracture is similar to the slot width or the width of the ow- permitting area during injection. Similarly, when considering slot
widths of 0.5, 0.25, 0.125, and 0.09 in., the resulting fracture heights are identical for each respective width. The core is anked by metal
sheets at both of its edges, as depicted in Fig. 7.
The pressure drop has been recorded at regular intervals of 1 minute, and the point of stabilization of the pressure drop for a specic
ow rate is continuously monitored. The pressure drop proles of the nano- silica solution ow at a concentration of 30% for varying
fracture heights are presented in Fig. 7. The presented gures demonstrate that, holding the fracture/crack width constant, the pressure
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2024 SPE Journal4
drop across the fracture/crack augmented as the ow rate increased. This phenomenon is ascribed to the ow friction. The data addition-
ally demonstrate that, for a consistent ow rate, a dilation in fracture or crack width corresponded to a decrease in pressure drop. This
phenomenon can be anticipated as the widening of fracture or crack leads to an expansion in the ow area, consequently resulting in a
decrease in friction loss. As per the Hagen- Poiseuille correlation, which applies to the laminar ow of Newtonian uids in tubes, an
increase in the volumetric ow rate will result in a corresponding increase in the pressure gradient. Moreover, augmenting the fracture
dimensions, namely, fracture widths (w) and fracture heights (h), which also implies wider slot width, can alleviate ow restrictions and
consequently result in reduced pressure drop.
Discussion
The experimental results are compared with the results of Guo et al.’s (2022) mathematical model. This model takes the following form:
Fig. 2—Schematic diagram of the experimental setup.
Fig. 3—Laboratory experimental setup.
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2024 SPE Journal 5
Fig. 4—Crack preparation using the metal sheet: (a)Top view; (b)side view.
Fig. 5—Cracked core jointed together using tape before placing it into the core holder.
Fig. 6—Nano- silica solution.
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2024 SPE Journal6
dp
dz
=
12
av
w
2
h+w
h,
(1)
where p is the pressure, z is the crack length, μ is the uid viscosity, h is the crack height, and vav is the mean ow velocity dened by
av =
Q
A
=
Q
wh ,
(2)
where Q is the volumetric ow rate of the uid, A is the cross- sectional area of the crack, and w is the average width of the crack of the
rectangular cross section. The variables in Eqs. 1 and 2 are in consistent units and converting from consistent units to U.S. oileld units
of psi/ft, cp, ft/sec, and in., we obtain:
dp
dz
=
av
4000w
2
h+w
h.
(3)
This mathematical model is used to calculate the pressure gradient for dierent conditions (varying “w,” “h,” “q,” and “c”) similar to that
used in the laboratory tests. The results are then compared with real- test data to see the uctuation of the results. The dierences between
test- derived pressure gradients and the model- predicted pressure gradients are around 93–99% in each case where the model- predicted
values are literally very small. It was rst hypothesized that the lack of account for NP- NP and NP- wall frictions, as well as the impact of
sandstone surface roughness, in the model’s development by Guo et al. (2022) was the reason for this disparity. The ratio between the test
data and the model- predicted data is estimated and plotted against four key parameters varied in this research (“w,” “h,” “q,” and “c”).
Their responses are shown in Figs. 5 through 8, and a correlation is formulated to modify Guo et al.’s (2022) model.
The plot depicted in Fig. 8 showcases the relationship between the ratio and fracture width for varying fracture heights, specically in
the context of a 5% concentration and diverse ow rates. Similar plots are produced for all other concentrations, though for the sake of
Fig. 7—Pressure drop profiles for 30% nano- silica concentration.
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2024 SPE Journal 7
brevity, only one instance has been illustrated herein. The graphical representations illustrate that there is a positive correlation between
the dimensions of fracture width (w) and height (h) and the ratio between the predicted data of the model and the actual test data. The
steepness of the plot with an increased fracture height is more pronounced in comparison to that of a reduced fracture height. This indi-
cates that there is a positive correlation between the size of fracture dimensions (“w” and “h”) and the magnitude of the disparities
observed between the test result and the model- predicted result. The proles indicate that the trend lines exhibit a near- linear pattern with
a low degree of steepness and a close correspondence between them for fracture heights of 0.125 and 0.09 in. The potential explanation
for this phenomenon could be attributed to the straightforwardness of the postulations underpinning the mathematical framework devel-
oped by Guo et al. (2022) and the experimental data obtained in controlled laboratory settings. Fig. 7 displays authentic test data that
exhibit distinct variations in results across dierent conditions. However, the predicted results from the model are considerably lower,
prompting a regression correlation analysis to be conducted subsequently in this study.
Fig. 8—Ratio vs. fracture width plot for different fracture heights in 5% concentration when (a) flow rate, q = 5mL/min, (b) flow rate,
q = 7.5mL/min, (c) flow rate, q = 10mL/min, and (d) flowrate, q = 12.5mL/min.
The plots depicted in Fig. 9 showcase the relationship between the ratio and fracture height for varying fracture widths, in the
scenario of a 5% concentration, across dierent ow rates. Similar plots are produced for all other concentrations; however, for the
sake of brevity, only one instance has been illustrated herein. The plot illustrates that the proportion of test data to model predicted
data rises in tandem with increases in both fracture width (w) and height (h), exhibiting a nearly proportional trend. Greater dispari-
ties between the predicted results of a model and the actual test results are indicated by higher ratios, particularly for larger values of
“h” and “w.”
The plot depicted in Fig. 10 displays the relationship between the ratio and ow rate for varying fracture widths, at a concentration of
5%, and across dierent fracture heights. The plot illustrates that the ratio between the test data and the model- predicted data remains
consistent, regardless of the ow rates, for fracture widths of 0.03 and 0.06 in. However, discrepancies are noted in instances where the
value of w is 0.09 in., which is the most slender among the three. This observed pattern has been consistently present across all instances,
encompassing ve distinct fracture elevations. Although the patterns are not identical, there is a slight increase in the ratio for higher ow
rates when the fracture width is very narrow.
The sensitivity analysis of the ratio between the test and model- derived data with the concentration of nano- silica solutions is depicted
in Fig. 11. The observed trend indicates a nearly linear increase in the values of “Ratio” as the concentration increases. The analysis
reveals that the disparity between the outcomes of the test and the data predicted by the model increases with the increase in the concen-
tration of the solution.
Extensive tests are conducted utilizing four ow rates for the amalgamation of three varied fracture widths and ve discrete fracture
heights. The nano- silica solutions exhibited a range of concentrations, with the least concentrated solution being 5% and the most con-
centrated solution being 30%. This results in a substantial collection of empirical data that is juxtaposed with the model proposed by Guo
et al. (2022). The observed disparities exhibit a signicant magnitude, resulting in an elevated “ratio.” Furthermore, there exists an insuf-
cient correlation that can be established to determine the impact of the manipulated parameters in this study. A theoretical relationship
among various parameters utilized in tests is established through the formulation of a nonlinear regression. The equation can be repre-
sented as follows:
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2024 SPE Journal8
Ratio = awb
1
hb
2
cb
4
,
(4)
where a, b1, b2, b3, and b4 = constants; Ratio = ratio between the test data and the model- predicted data; w = fracture width, in.; h = fracture
height, in.; q = ow rate, mL/min; c = concentration, %.
Eq. 4 is linearized using logarithm on both sides to give
Fig. 9—Ratio vs. fracture height plot for different fracture widths in 5% concentration when (a) flow rate, q = 5mL/min, (b) flow rate,
q = 7.5mL/min, (c) flow rate, q = 10mL/min, and (d) flow rate, q = 12.5mL/min.
Fig. 10—Ratio vs. flow rate plot for different fracture widths in 5% concentration when (a) fracture height, h = 1in., (b) fracture
height, h = 0.5in., (c) fracture height, h = 0.25in., (d) fracture height, h = 0.125in., and (e) fracture height, h = 0.09in.
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2024 SPE Journal 9
log(Ratio) = log(a) + log (w)b
1
+ log (h)b
2
+ log (q)b
3
+ log (c)b
4
,
or
log(Ratio) = log(a)+b1log (w)+b2log (h)+b3log (q)+b4log (c).
For simplication we assume
Y= log(Ratio); A= log(a); W= log(w);
H= log(h); Q= log(q); C= log(c).
The new equation takes the form of
Y=A+b1W+b2H+b3Q+b4C.
(5)
Solving with the test data using regression in Excel, the coecients and the intercepts are determined, and the equation becomes
Y= 5.4773 + 2.4308W+ 0.4713H0.05312Q+ 0.367023C,
(6)
Fig. 12—Relationship between actual ratio and correlation ratio of the test data and model- derived data.
Fig. 11—Ratio vs. fluid concentration plot for different fracture widths (1 in. height) when (a) flow rate, q = 5mL/min, (b) flow rate,
q = 7.5mL/min, (c) flow rate, q = 10mL/min, and (d) flow rate, q = 12.5mL/min.
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2024 SPE Journal10
where
A= log(a) = 5.4773; a= 300123.50,
b1= 2.4308; b2= 0.4713; b3=0.05312; b4= 0.367023.
Based on Eq. 6, the logarithmic function
Y= log(Ratio)
was utilized to estimate the value of “R” across all pertinent conditions associated
with the injection test scenario, including variations in “w,” “h,” “q,” and “c.” The aforementioned values are commonly referred to as
the “Correlated Ratio” in academic discourse. In Fig. 12, we depict the plotted correlated ratio in comparison with the “Actual ratio,”
which is derived from the comparison between the test result and the model- predicted result. The visual representation portrays a strong
correlation among the data points, with an R2 value of approximately 90%, and a tendency to converge towards the trendline. The high R2
value indicates that the linear regression model for the correlated ratio is a good t for the data set derived from the test results and math-
ematical model. The study’s usage of nonlinear regression correlation, model- derived data, and test results serves to substantiate its
credibility.
Conclusions
This study conducted an experimental investigation to analyze the ow behavior of nano- silica solutions within ducts possessing rectan-
gular cross- sectional geometries. Various fracture dimensions (height and width) are used where the concentrations of the solutions are
varied too. The basic work is almost similar to the previous two works mentioned in this article but has been extended toward using a uid
of dierent concentrations and incorporating the regression analysis to ascertain the credibility of the method and the data obtained. Based
on this assessment, it can be deduced that the following conclusions can be drawn:
1. The nano- silica solution pressure drops in conduits of narrow cross sections over a given length cannot be described by the existing
hydraulics models found in the literature. The ratio of the measured pressure drop value to the model- predicted value depends on uid
ow rate, geometry of the cross- sectional area, and nano- silica concentration.
2. An empirical relation was developed to quantify the ratio as a function of nano- silica solution ow rate, cross- section geometry param-
eters, and nano- silica concentration based on nonlinear regression analysis. The empirical relation highly correlates to the actual ratio
with a correlation coecient of R2 = 0.8965.
3. In real applications, the pressure drops given by Guo et al.’s (2022) hydraulics model should be multiplied by the ratio given by the
empirical correlation to minimize error.
Data Availability Statement
The data used in this research is condential. It can only be released with the permission of the appropriate authority upon request.
Acknowledgments
The authors would like to express their gratitude to the University of Louisiana at Lafayette. Their extreme appreciation to BIRD for their
support to the project "Safe, sustainable, and resilient development of oshore reservoirs and natural gas upgrading through innovative
science and technology: GOM—Mediterranean" under Contract No. EC- 19 Fossil Energy.
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