Fine Migration Control in Sandstones: Surface Force Analysis and Application of DLVO Theory.

Formation damage caused by fine migration and straining is a well-documented phenomenon in sandstone reservoirs. Fine migration and the associated permeability decline have been observed in various experimental studies, and this phenomenon has been broadly explained by the analysis of surface forces between fines and sand grains. The Derjaguin-Landau-Verwey-Overbeek (DLVO) theory is a useful tool to help understand and model the fine release, migration, and control phenomena within porous media by quantifying the total interaction energy of the fine-brine-rock (FBR) system. Fine migration is mainly caused by changes in the attractive and repulsive surface forces, which are triggered by mud invasion during drilling activity, the utilization of completion fluid, acidizing treatment, and water injection into the reservoir during secondary and tertiary recovery operations. Increasing pH and decreasing water salinity collectively affect the attractive and repulsive forces and, at a specific value of pH, and critical salt concentration (CSC), the total interaction energy of the FBR system (V T) shifts from negative to positive, indicating the initiation of fine release. Maintaining the system pH, setting the salinity above the CSC, tuning the ionic composition of injected water, and using nanoparticles (NPs) are practical options to control fine migration. DLVO modeling elucidates the total interaction energy between fines and sand grains based on the calculation of surface forces of the system. In this context, zeta potential is an important indicator of an increase or decrease in repulsive forces. Using available data, two correlations have been developed to calculate the zeta potential for sandstone reservoirs in high- and low-salinity environments and validated with experimental values. Based on surface force analysis, the CSC is predicted by the DLVO model; it is in close agreement with the experimental value from the literature. The critical pH value is also estimated for alkaline flooding. Model results confirm that the application of NPs and the presence of divalent ions increase the attractive force and help to mitigate the fine migration problem. Hence, a new insight into the analysis of quantified surface forces is presented in current research work by the practical application of the DLVO theory to model fine migration initiation under the influence of injection water chemistry.

Introduction

The decrease in ionic strength (Is) and change in ionic composition of formation water lead to fine migration and permeability reduction in subsurface porous and permeable sandstone reservoirs containing various types of clay minerals. Such minerals are present in the pore space as either agglomerate particles or as fine particles which cover the sand grains. Clay minerals are alumino-silicates with a very specific layered structure. The basic building blocks of clay minerals are layers of silica, alumina, and magnesia. There are major three types of clay minerals which include kaolinite, montmorillonite, and illite/mica.[1,2] Sandstones consist of a matrix of quartz grains enclosing an interconnected pore space. The nature of the depositional environment can lead to the presence of siliceous fines because of grain compaction and crushing because of applied stresses as well as fine deposition. These fines are generally held within the formation water film that surrounds the quartz grain in water-wet conditions.[3] Fine migration may happen in different types of natural and technical processes, such as water aquifer recharging by some external water source, underground formation water disposal, groundwater flows, and invasion of drilling muds, invasion of completion fluids, acidizing and waterflooding treatments, high rate oil and gas production and injection, and improper design of oil recovery processes with low-salinity water injection.[4−15]

Water injection operations into sandstone and carbonate reservoirs performed by reducing the salinity and tuning the ionic composition are a promising and evolving technology to maximize oil recovery, primarily by modifying the wettability of the crude–brine–rock system.[16−25] During the aforementioned process of waterflooding, the salinity, chemistry, and injection rate of injected brine play a vital role in altering the rock wettability and changing the surface forces between fines and sand grains in sandstone reservoirs, which affect the efficiency of the procedure.[17,24−30] Fine particles can detach, become suspended in the injected fluid, and form a colloidal system in the reservoir because of the alteration of attractive and repulsive surface forces; while they move with the injected fluid/brine, they may block the pore throats. This phenomenon is referred to as straining: it blocks already open pores and results in formation damage, with a substantial decline in formation permeability,[5,31−34] as shown in Figure .

Figure 1

Migration of natural fine particles in the reservoir (reproduced with permission from J. Nat. Gas Sci. Eng.2020,73, 103047).[34]

Fine migration in sandstone reservoirs is supposed to be one of the possible mechanisms of enhanced oil recovery (EOR) in low-salinity projects. It provides better mobility control by plugging some of the pores, diverting flow toward unswept sections of the reservoir, and increasing the sweep efficiency, which eventually is favorable for incremental oil recovery.[2,27,28,35−42] As yet, fine-assisted oil recovery phenomenon is still not well understood and in-depth research on this issue is necessary to address the associated productivity and injectivity decline. On the other hand, fine migration has been reported by some researchers to have adverse effects on fluid productivity and injectivity, and their release, migration, and straining can significantly impair the hydraulic connectivity of the reservoir because they plug the actual path for fluid flow.[43−47]

Fine release and migration inside sandstone reservoirs are related to several factors such as the available concentration of fine particles, ionic composition, and salinity of injected water, pH, wettability, flow rate, and relative flow of different phases.[7,12,48,49] In some recent studies,[50−55] it has been found that CO2 injection in sandstone can also lead to permeability reduction because of fine migration. Mineral dissolution caused by CO2–brine–rock interaction could be the reason for fine particle generation in porous media and the related decline in CO2 injectivity. Therefore, all these key factors must be considered to get an in-depth understanding of interactive forces between fine particles and sand grains in order to properly design a solution and mitigate the fine migration problem.

Various techniques have been developed and utilized to overcome the aforementioned problems and enhance oil recovery economically. Some examples include the utilization of clay stabilizers, matrix acidizing treatment,[56,57] adjusting the salinity, tuning the ionic composition, and changing the pH of the injected water.[30,58,59] The application of nanoparticles (NPs) is one of the emerging technologies used to fix the fine migration problem. NPs are extremely small particles; their size can vary between 1 and 100 nm, and they have a high surface-area-to-volume ratio because of their small size. NPs can change the surface properties of the materials to which they are adsorbed. A single type of NPs [in the form of nanofluids (NFs)], a combination of more than one type of NP (hybrid case), and also NPs and surfactants in combination are being used to reduce formation damage and enhance oil production.[60−65] NPs alter surface forces and potential as they are adsorbed on the rock. Several types of NPs with different chemical natures and distinct properties have been used to control fines migration; they include magnesium oxide (MgO),[61,66−68] silicon oxide (SiO2),[60,69] and aluminum oxide (Al2O3).[65,69−71]

In the past, Martin[72] performed many waterflooding experiments on sandstone cores and observed that a decrease in injected water salinity resulted in additional oil recovery, accompanied by a decrease in core permeability because of clay swelling. Later, Khilar and Fogler[73] found that when injected fluid salinity falls below a critical salt concentration (CSC), fines are released and migration starts within porous media. Fine migration and subsequent permeability reduction were also confirmed by experimental research.[7,27,35,36,50,74−76] Kumar et al.[22] and Mansouri et al.[60] used scanning electron microscopy (SEM), the field emission SEM, and atomic force microscopy to visually show the mobilization of mixed-wet kaolinite particles with high-resolution images.

Attractive forces are responsible for retaining fines on the rock surface, whereas repulsive forces try to detach fines and promote migration. The Derjaguin–Landau–Verwey–Overbeek (DLVO) model incorporates surface forces and calculates the total interaction energy for the system, which is either positive or negative based on the contribution of each energy component.

In this paper, surface forces are quantified, and the DLVO model is used to predict the CSC for NaCl and to estimate a critical pH value for alkaline flooding. The model confirms that NPs increase the attractive energy and help fixate fines on sand grains. Furthermore, this tool has estimated an even lower CSC if there are divalent ions in the solution, which suggests the idea of tuning/adjusting the ion composition of injection water to avoid fine migration. Two correlations for the zeta potential calculation have also been developed and validated using further modeling. Therefore, the present study concludes that the analysis of quantified surface forces combined with DLVO modeling is a powerful tool to predict and control fine migration in porous media, and further research on the sensitivity of important parameters can improve the results.

Fine–Brine–Rock System

The detachment of fine particles from the sand grain surface is the initial step in the process of fine migration in sandstones. A comprehensive understanding of this detachment process is necessary to analyze conditions for the migration and the resultant formation damage. Generally, two types of forces are responsible for the detachment and mobilization of fine particles. These forces are classified as colloidal forces and hydrodynamic forces. Colloidal forces are electrostatic in nature, and they are further divided into two types, which are London–van der Waals attractive forces and electrical double-layer repulsive forces between particles and surfaces. The hydrodynamic forces are mainly related to the flow of permeating fluid through porous media. Synthetic fines and silica glass beads have been used in numerous studies,[60,65−67,69,71,77] to mimic sandstone reservoirs with fine particles, as shown in Figure , because they are spherical. However, kaolinite in sandstone reservoirs has a platelet structure with a finite thickness, while natural sand and glass beads both have infinite thicknesses (IT) as compared to fine particle size. Based on the aforementioned configuration of synthetic/natural fines and sand/glass beads, generally, there are different electrostatic energies for two different systems: the sphere–IT plate and the kaolinite platelet–IT plate.

Figure 2

Adsorbed fines on the surface of a glass bead (reproduced with permission from Colloids Surf., A2013,436, 803–814).[65]

An analysis of a single fine particle of presumably spherical shape on a sand grain surface had been performed to describe the conditions required for the detachment of fine from a flat surface.[78−81]Figure describes the sphere–IT plate model designed to mimic a spherical fine particle attached to the pore/rock grain through which a high-salinity permeating liquid is flowing. However, Figure demonstrates a kaolinite platelets–IT plate configuration, where small kaolinite platelets are present on a sandstone grain. Most of these platelets are in the form of clusters and can move together based on attractive forces between individual plates. Similar kaolinite platelet configurations have also been found in other sources.[82−87]

Figure 3

Spherical fine on sand grain surface.

Figure 4

Kaolinite platelets–IT plate configuration.

The separation distance (h) between a fine particle and the pore surface in Figure is quite small (usually on the order of 10–1 nm), and additionally, these fine particles are subjected to the hydrodynamic forces of the flowing liquid during production and injection processes. There are different energy contributions from colloidal and hydrodynamic forces, and the total energy of all interactions between a fine particle and the pore/grain surface must be determined precisely in the DLVO model to incorporate the effects of attractive and repulsive forces. If the net interaction energy of the system comes out to be positive, it means repulsive forces have dominance over the attractive forces; as a result, fines will be detached from the surface, and migration will start in the porous medium.

During early research studies and to date, a single fine particle of spherical shape on a sand grain flat surface (sphere–plate model) has been extensively used for the calculation of DLVO interactions because of the simplicity of the approach.[33,65,67,71,77,88−92] A few researchers have utilized a plate–plate model for the quantification of interaction energies.[93,94] The single sphere model can be accurately used for synthetic fines and glass bead configurations, but when it comes to natural kaolinite and sand grain configurations, it can provide erroneous results because natural kaolinite has a platelet structure and must be modeled with a kaolinite platelets–IT plate model. In some studies during the last few years,[31,32,95−99] clustered fine particles’ detachment and combined movement were assumed instead of a single fine particle model. Recently, Chequer et al.[90] used this new idea to show that the single-colloid single-surface system is not an accurate representation of colloidal behavior in porous media and significantly underestimates the critical velocity of the fluid to initiate the fine migration. Experimental results were in close agreement with the clustered fines model.

DLVO Theory

The well-known DLVO theory was established by Derjaguin, Landau, Verwey, and Overbeek.[100−102] It describes the VT of the system incorporating attractive and repulsive forces because of the van der Waals attractive potential (VLVW), electric double-layer (EDL) potential (VEDL), and Born repulsive potential (VBR). This theory assumes that the VLVW, VEDL, and VBR potentials are independent of each other and therefore can be added for the particle–plate system, using either a sphere–plate or plate–plate model configuration, to quantify total interaction energy at each interacting distance. The DLVO-based VT of the system composed of a fine particle and a pore surface is presented as eq .

Generally, the DLVO theory provides good estimates for the surface–surface forces with a separation distance of around 5 nm, provided that all the important parameters, such as the particle size, Is, Hamaker constant (AH), and zeta potential (ζ), are accurately and precisely measured. The total energy of a specific system can range from positive (repulsive) to negative (attractive) depending on the individual contributions of the attractive and repulsive forces.

Application of DLVO Theory in the Petroleum Industry

The DLVO theory has been widely used in the petroleum industry to quantify surface forces between fines and sand grains during NF injection scenarios, disjoining pressure estimation, and polymer and surfactant adsorption on the rock surface. Quantification of DLVO-based interactions has been reported to be in close agreement with experimental results. Habibi et al.[71] utilized NPs in synthetic cores to mitigate the fine migration problem, computed total interactions, and found that MgO NPs improved the attractive forces between fines and the grain surface. Arab and Pourafshary[65] performed several experiments on glass beads to mitigate the fine migration issue, accompanied by low-salinity flooding. They used five different types of NPs to control fine migration and found that ZnO and γ-Al2O3 NPs were the best at this task. In addition to experimental results, they also measured the zeta potentials of the system before and after the application of NFs and applied a DLVO sphere–IT plate model to calculate interactive energy. The total energy was attractive after the application of NPs as compared to the nontreated case. Arab et al.[67] performed experiments using SiO2 and MgO NPs in the preflush mode and found that 0.03% MgO NPs performed the best among all scenarios. They confirmed their experimental findings with the DLVO theory by calculating the total interaction potential between the rock and fine particles.

Assef et al.[66] demonstrated their work to mitigate colloidal particle movement in porous media by using MgO NPs, and 97% retention of fines was observed. They utilized the extended DLVO (X-DLVO) theory by incorporating the effect of acid–base energy and neglecting the hydrodynamic forces and quantified the total interaction energy of the system. Zou et al.[103] applied the X-DLVO theory to investigate the adsorption of anionic polyacrylamide onto coal and kaolinite particles. Based on the results, they observed that the VT between kaolinite and coal particles was repulsive after the adsorption of the polymer on coal particles, which proves the effectiveness of the mechanism of coal purification.

Quantification of Interaction Energies

London–Van Der Waals (VLVW) Interaction Energy

In particle physics, there exists an attractive force between similar particles/plates when they are infinitesimally close to each other. A German-American physicist, Fritz London, published the first satisfactory microscopic theory of dipole–dipole dispersion forces.[104] This attractive force is a distance-dependent force between molecules, atoms, and particles and does not have any association with any type of ionic or covalent bonds. It decays slowly and acts at a distance less than 10 nm. The main cause of this electrostatic force is the presence of permanent and oscillating dipoles of atoms.[12] These forces are weak chemical forces, but still play a critical role when colloidal particles are infinitesimally close to each other in a solution. Based on the sphere–IT plate model, London–van der Waals energy (VLVW) is presented in two forms, in eqs ,[79] and 3.[105]

Chequer et al.[90] presented a new model based on the clustered fine sphere–IT plate model, as shown in eq .

On the other hand, a separate model for kaolinite platelets–IT-plate configuration has been presented by Gregory et al.,[79] as shown in eq .

The negative sign of VLVW demonstrates the attractive nature of this potential.

EDL (VEDL) Interaction Energy

When charged colloidal particles (fines) are immersed in an electrolyte solution of specific ionic strength, mobile ions from the electrolyte solution form an ionic film around the particles.[102] Based on the positive or negative charge of a particle, oppositely charged ions from the surrounding electrolyte solution are attracted and form an ionic layer over the charged particle called a compact layer (stern layer), which is moved with the particle. The excess charge on the compact layer is balanced by the oppositely charged ions from the electrolyte solution forming another layer, which is called the diffuse layer (slipping plane). In the diffuse layer, ions are not tightly bound to each other and are free to move to and from the electrolyte solution. These two layers are electrostatic, and their combined effect is called the EDL, as shown in Figure .[106]

Figure 5

Compact and diffuse EDLs and corresponding zeta potential (reproduced with permission from Elsevier: Amsterdam, The Netherlands, 1995; pp 113–156).[106]

The potential difference between these two layers is called zeta potential and is denoted by ζ. Zeta potential provides the closest surface potential estimate and is used in the quantification of the EDL interaction energy. Zeta potential is not directly measured and is obtained by applying an electric field across the dispersion: this process is called electrophoresis.[107,108] Particles within the dispersion with a specific zeta potential value move toward the electrode of opposite charge with a velocity proportional to the magnitude of the zeta potential. At lower ion strength, such as in the low-salinity injection condition, EDLs that have already formed around the sand grain surface and the fine particles expand and overlap, which leads to a repulsive interaction energy (VEDL). High repulsive force may detach the fine particles from the sand grain surface. The repulsive force is higher at a lower solution salinity.

Regarding the formulation of VEDL, the simplest case of sphere–plate geometry is used, as shown in Figure . For the boundary conditions, the rock and fine particle surfaces both may have a constant surface potential or constant charge, or one of the surfaces may maintain its charge density constant while the other surface possesses a constant potential (mixed case). Generally, fine migration can be modeled with a constant potential case because zeta potential is easy to measure as compared to surface charge. VEDL can be calculated by different formulae, such as eqs , 7,[109] or 8,[80] for the sphere–IT plate model.

Equation is valid for potentials less than 60 mV, when the double-layer thickness is less than the fine particle size, which is true in most scenarios of fine migration in sandstone porous media. Ψ1 and Ψ2 are the surface and compact layer potentials, respectively, and can be replaced by the measured value of the zeta potential (ζ-potential) to develop eq . Chequer et al.[90] presented a new model to calculate VEDL based on the clustered fines sphere–IT plate model, as shown in eq .

For VEDL calculations based on the kaolinite platelets–IT-plate model, Gregory[110] presented a different model, as shown in eq .

Born Repulsion (VBR) Interaction Energy

In a colloidal system, when particles approach and are about to contact each other, a short-range repulsive potential called the Born repulsion potential (VBR) is generated because their electron clouds overlap. This potential is quite sensitive to the structure of surfaces in contact and permeating liquid. Formulations to quantify the Born repulsion potential for the previously described sphere–IT-plate system have been presented by Ruckenstein et al. and Schumacher et al.[111,112] in eq and Mahmood et al.[81] in eq .

For clustered fine movement in the porous medium, Chequer et al.[90] presented a new model to calculate VBR, as shown in eq .

A separate model for the natural kaolinite platelets–IT-plate configuration has been presented by Mahmood et al.[81] as eq .

To compute Born repulsion accurately for the fines–rock–fluids system configuration, AH and σ (atomic collision diameter, nm) must be known precisely. An average value used for σ in the calculation of VBR is around 0.5 nm. Generally, VBR has a very small impact on the VT and can be neglected in comparison to other electrostatic potentials (VLVW and VEDL) if the separation distance is greater than 1 nm.

Results and Discussion

The EDL expands because of alteration in the pH and ionic strength of the solution, which affects the repulsive force. In this section, we study the effect of these parameters on the total force and fine migration. As mentioned before, the application of NPs increases attractive energy by changing the surface potential to control fine migration, which is also investigated, and results are shown in this section using the DLVO approach.

pH of Solution

The pH of colloidal dispersions is one of the most important factors that affect the repulsive force, and it is indicated by the change in the zeta potential of the system. Zeta potential is generally positive at low pH values (acidic region), and with increasing pH, it becomes negative because of the presence of excess OH–1. There is a specific pH, where the zeta potential becomes zero, which is called the point of zero charge (PZC) or the isoelectric point (IEP), as shown in Figure . For a pH higher than the PZC, the surface charge becomes negative, which means a repulsive force that leads to the separation of fines. NPs can shift the PZC by changing the surface forces, so they can be used to control the repulsive force and the detachment of fines. The PZC for SiO2 NPs is around pH = 2.5–3. For alumina, it lies between 7.5 and 9, and for MgO, it is around 12–13.[113] Hence, for highly alkaline conditions, the application of MgO NPs prevents fine detachment even at high pH values.

Figure 6

Change in zeta potential with pH of the solution (reproduced with permission from Woodhead Publishing, 2016; pp 299–325).[114]

pH Sensitivity Analysis Based on the DLVO Model

During alkaline flooding EOR, the alkali generates in situ surfactants that reduce the oil–water interfacial tension to maximize oil recovery. However, alkaline flooding in sandstones may cause fine migration problem because of the change in surface potential caused by an alteration in pH. As the pH of the system increases during alkaline flooding, the repulsive force between fine particles and sand grains increases, and the corresponding zeta potential decreases (becomes more negative) because of excess OH–1 in the system. This mechanism results in fine detachment and migration during alkaline flooding. The changes in zeta potential caused by increasing the pH of a high-salinity system, composed of a 2 M solution of 8 wt % NaCl with 2 wt % CaCl2 and crushed sandstone grains with kaolinite particles, were measured and shown in Table . The DLVO model is used to calculate total energy for the case shown in this table and analyze the effect of pH values ranging from 2 to 8.5, as shown in Figure . Surface forces have been quantified, and the DLVO model has been applied to determine a critical pH value above which fine migration may begin.

Table 1

Zeta Potential Data Based on pH of the System

ionic strength	pH	ζ-potential (mV)	source
2 M (8 wt % NaCl + 2 wt % CaCl2)	2	–7	Singh and Mohanty[115]
	3.5	–11
	5	–16
	6	–22.5
	6.5	–26
	7	–28
	8.5	–32

Figure 7

DLVO interactions and pH sensitivity analysis.

The critical pH value to initiate fine migration is between 6 and 6.5 for this case. Thus, maintaining the system pH below 6 can prevent fine migration. This calculation shows that the application of alkaline flooding leads to fine migration because in alkaline flooding the pH will be more than 7.

Ionic Strength (Is)

The ionic strength of the solution affects the expansion and thickness of the EDL, which further affects the repulsive force and total energy of the system. The higher the ionic strength of the solution, the more compressed the EDL becomes. For the quantification of EDL repulsion, the ionic strength and the corresponding experimental zeta potential data have been taken from research work by Chequer et al.[90] The effect of decreasing the ionic strength on the Debye length (k), which is an indicator of EDL thickness, is shown in Table . As the solution salinity decreases, EDLs around the fine particles and the sand grains both expand simultaneously, generating more repulsion between the fines and the sand grains. Consequently, detachment of the fines occurs.

Table 2

Effect of Ionic Strength on EDL Thickness

NaCl Is at 298 K (M)	(nm)	ζ-potential (mV)[90]
0.6	0.4	–17.9
0.4	0.5	–21.3
0.2	0.7	–24.3
0.1	1	–30.5
0.05	1.4	–33.7
0.025	2	–34.0
0.01	3	–34.3

The low zeta potential confirms the high repulsive force in these conditions. Figure demonstrates the increase in repulsive force when the ionic strength is reduced from a high-salinity (0.6 M) to low-salinity environment (0.01 M). However, a decrease in brine salinity does not affect van der Waals attraction at any molarity, as depicted by the DLVO model in Figure a. Both energy contributions result in the total energy for the system incorporating constant attraction and variable repulsion at different salinities, indicating a critical salinity from 0.2 to 0.4 M, below which we may have fine detachment, as shown in Figure b.

Figure 8

Effect of ionic strength on the EDL repulsion.

Figure 9

Effect of ionic strength on (a) attractive force and (b) total interaction energy.

For systems containing silica glass beads, fines, and 0.02 M NaCl, CaCl2, and MgCl2 solutions, zeta potentials have been measured by Assef et al.[66] Data for 0.3–0.1 M NaCl have been measured by Chequer et al., as shown in Table . We applied the DLVO approach to estimate the CSC value for these cases. Calculations of attractive and repulsive forces have been made for salinities from 0.3 to 0.02 M, and the results are shown in terms of the dimensionless VT of the system. Data used in the model are given in Table .

Table 3

Zeta Potential Data for Analysis

salt	Is (M)	ζ-potential (mV)	source
NaCl	0.3	–16	Chequer et al.[90]
	0.25	–18
	0.2	–19
	0.1	–23
	0.02	–24.3	Assef et al.[66]
CaCl2	0.02	–8.3
MgCl2	0.02	–6.09

Table 4

Constants

parameter	symbol	value
fine particle radius	ap	5 × 10–7 m
fluid temperature	T	297.15 K
pH of solution	pH	7
Boltzmann constant	kB	1.38 × 10–23 J K–1
Hamaker constant	AH	3 × 10–21 J
permittivity of free space	eo	8.85 × 10–12 C2 J–1 m–1
electron charge	e	1.6 × 10–19 C
Avogadro’s number	NA	6.02 × 1023 mol–1
pi	π	3.1416
atomic collision diameter	σ	0.5 nm

Figure shows the effect of changing the salinity from 0.3 to 0.02 M NaCl solutions containing silica glass beads and fine particles to mimic a sandstone formation. The total DLVO energy for each scenario has been calculated using the sphere–IT plate DLVO model. As shown in Figure , total interaction energy is attractive for the high-salinity case because of the weak EDL repulsion. As the salinity is gradually decreased from 0.3 to 0.2 M, total system energy shifts from negative to positive, indicating that the EDLs have expanded and overlapped, causing an increase in repulsive forces (which dominated the attractive forces) under low-salinity conditions. It is observed from the DLVO model calculation that the CSC for the NaCl system is in the range of 0.2–0.25 M, which is in close agreement with the experimental value for the NaCl solution.[12,60]

Figure 10

Prediction of CSC through DLVO interactions.

Nanoparticles

Data concerning various NPs used in different studies, which were performed for fine fixation, wettability modification, interfacial tension (IFT) reduction, drilling mud preparation, and stable NFs formulation, have been collected, and the results have been presented in Figure .

Figure 11

Different NP utilization in no. of studies.

It is observed that the maximum number of studies are performed using SiO2 NPs because of their high stability, low toxicity, resistance to pH changes, and easy availability.[116] MgO and Al2O3 NPs are next on the list and have been utilized in several studies to mitigate fine release, the migration issue, and IFT reduction, respectively. NPs are used to overcome the fine migration problem in sandstone reservoirs by changing the surface forces, which can be observed by an alteration in zeta potential values before and after the adsorption of NPs onto rock/grain surfaces. Figure presents zeta potential values of glass beads before and after the application of different NPs in low-salinity conditions.[60,63,65−67,69,71,77,117]Figure shows that MgO NPs are the best at changing the surface energy and increasing zeta potential. In a few research studies, ZnO, TiO2, and NiO NPs, and combinations of more than one NP, have been used in the form of NFs. Future research will benefit most from using different types of single and hybrid NFs. In an experimental study, Assef et al.[66] used MgO NPs to mitigate the adverse effects of fine migration that were noticed under low-salinity flow conditions (0.02 M NaCl).

Figure 12

Change in zeta potential by the application of NPs.

They observed experimentally that the application of only 0.0075 wt % MgO NPs retained around 97% of in situ fine particles even in highly alkaline conditions of pH = 9.2. Retention of fine particles was attributed to increasing attractive forces between fines and glass beads in the presence of MgO NPs. To validate the experimental results, we applied the DLVO theory to quantify the total energy of the system, which is presented in Figure . Surface force quantification and analysis showed that the application of NPs increased the attractive forces between fines and grains and consequently the VT of the system decreased, indicating a shift from repulsion to attraction.

Figure 13

DLVO interactions for MgO NPs in NaCl solution.

Divalent Ions

The presence of divalent ions (Ca2+, Mg2+) in the solution is beneficial and can suppress the EDL thickness, resulting in reduced repulsive force as compared to monovalent ions (Na+) for the same salinity. Xie et al.[118] performed experimental research and measured zeta potentials for a solution containing crushed sand particles in the presence of monovalent (Na+) and divalent ions (Ca2+, Mg2+) separately. They observed that divalent ions suppressed the EDL which was formed around the sand grain and hence resulted in reduced repulsion, which is indicated by low zeta potential (absolute value), as shown in Figure .

Figure 14

Monovalent and divalent ions effect on ζ-potential (reproduced with permission from J. Mol. Liq.2016,221, 658–665).[118]

Similar results for zeta potential data were observed in the studies performed by Shehata and Nasr-El-Din[119] and Gulgonul[120] on sandstone and natural hydrophobic Teflon, respectively, in the presence of monovalent and divalent ions (Ca2+, Mg2+). Assef et al.[66] also utilized CaCl2 and MgCl2 solutions and performed similar experiments. Quantification of surface forces has been done, and the DLVO model has been applied. Figure depicts the relative comparison of VT for monovalent and divalent ions in the electrolyte solution. The presence of divalent ions reduces the repulsive force and suppresses the expansion of the EDL even under low-salinity conditions because of their high valence charges. Divalent ions in the solution can help to design a lower value of the CSC, which means that low-salinity flooding benefits can be achieved without fine migration in sandstone reservoirs by tuning the composition of the injection brine.

Figure 15

Effect of divalent ions on DLVO interactions.

Interactive Parameters for the FBR System

Van der Waals particle–particle attraction is produced when particles/molecules come close to each other in a medium. The Hamaker constant is a coefficient that relates this interactive energy among particles, whereas zeta potential (ζ) is a measure of the surface charge of the particles and an indicator of increase and decrease of the repulsive force. The following section discusses these parameters in detail.

Hamaker Constant (AH)

Hamaker constant is dependent on the integrated system of the fine particle, its shape, pore surface, aqueous medium type, and salinity, and crude oil properties.[80] The Hamaker constant is determined experimentally with great caution based on the specific system configuration. The typical values for AH are found in the range of 10–21 to 10–19 J. These experimental values are in close agreement with the theoretical calculations of Israelachvili.[80,121]Equation is used to calculate this constant using experimental data.ε is the static dielectric constant, η is a refractive index, the subscripts 1, 2, and 3 refer to particles, grains, and water/brine, respectively, hp is the Planck constant (6.62 × 10–34 J s), and νe is a constant value of electronic adsorption frequency equal to 3 × 10–15 s–1. Table presents studies where Hamaker constants were calculated and a few studies where experimental approaches have been used to measure AH based on the system’s actual configuration.[90,93,122,123] A sensitivity analysis for AH has been performed and total interaction energy has been quantified in Figure . A high Hamaker constant leads to more attraction, whereas its low value is related to reduced attraction, as demonstrated by the DLVO model in Figure a. Also, particle deposition onto a rock’s surface is affected by the Hamaker constant of the interactive system. Elimelech et al.[124] found that at low salinity, the range of EDL is much higher than that of van der Waals attraction, and the rate of particle deposition is mainly controlled by double-layer repulsive energy.

Table 5

Hamaker Constant Data

author	year	base liquid	pH	T (°C)	system	AH (J)	experimental/theoretical
El-Monier and Nasr-El-Din[93]	2011	distilled water	12	149	kaolinite–quartz	1.61 × 10–20	experimental
Habibi et al.[71]	2012	water	7	25	glass beads–water	6 × 10–21	theoretical
El Badawy et al.[122]	2012	0.03 M NaCl	7		metallic NPs	6.04 × 10–20	experimental
Arab and Pourafshary[65]	2013	water	7	25	glass beads–water	6 × 10–21	theoretical
Arab et al.[65]	2013	0.03 M NaCl	6.9	25	glass beads–brine	1 × 10–21	theoretical
Arab and Pourafshary[67]	2014	water	7	25	glass beads–water	6 × 10–21	theoretical
Habibi et al.[88]	2014	0.03 M NaCl	6.5–7	25	sand–NaCl	1 × 10–20	theoretical
Xie et al.[25]	2014	0.2 wt % NaCl	8.1	65	oil/silica in water	8 × 10–21	theoretical
Yang et al.[89]	2016	water		25	kaolinite and quartz	2 × 10–20	theoretical
Mahani et al.[21]	2017	0.035 M diluted sea water	7	25	limestone–brine	1 × 10–19	theoretical
Hasanneja et al.[77]	2017	0.3 M NaCl	7	25	glass beads–brine	1 × 10–20	theoretical
Xie et al.[125]	2018	NaCl, MgCl2 and CaCl2	4–10	25	shale-oil	0.81 × 10–20	theoretical
Huang et al.[91]	2018	2 wt % KCl	7	25	coal–brine	4.62 × 10–20	theoretical
Sanaei et al.[94]	2019	0.01 M NaCl	7	25	carbonate–brine	1.3 × 10–20	theoretical
Chequer et al.[90]	2019	0.6 M NaCl	7	25	kaolinite–sand	1.49 × 10–20	experimental
Takeya et al.[23]	2020	0.1 M ALSW	7	25	calcite–brine	6.6 × 10–21	theoretical
Tangparitkul et al.[33]	2020	0.0005 M NaCl brine			clay–sand	2 × 10–21	theoretical
Gomez-Flores et al.[92]	2020	0.001 M NaCl	7	25	silica–brine	3.91 × 10–21	theoretical
Peng et al.[123]	2020	0.007 M SDS and 0.0005 M NaCl		25	surfactant–water	5.2 × 10–20	experimental

Figure 16

Effect of Hamaker constant on (a) DLVO interaction energy and (b) particle deposition (reproduced with permission from Elsevier, 1995; pp 113–156).[124]

However, at high-salinity, van der Waals attractive forces are more effective, and as a result, the rate of particle deposition is controlled by AH. Figure b illustrates the effect of different AH values on particle deposition at different salinities. More particle deposition is observed because the repulsion is reduced at high AH in a high-salinity environment. Hence, it is recommended that the experimental value of Hamaker constant be used in modeling, for the fine particle, rock grain, temperature, and an aqueous medium that could be distilled water or brine of specific salinity, to get accurate results using the DLVO model.

Zeta Potential (ζ)

Zeta potential is an important indicator of surface charge and is used for the quantification of the electrostatic repulsive energy between dispersed fines and sand grains. It is the potential difference between the surfaces because of the electrical difference between a particle surface and points away from the particle in the fluid at the boundary of the slipping plane,[126,127] as shown in Figure . This potential is also known as electro-kinetic potential in colloidal dispersions. It is worth mentioning here that the measured zeta potential of clay particles (fines) is negative and is an important parameter that provides information about the charge on the particle surface, colloidal system stability in an ionic environment, electrostatic forces between particles and the rock surface, and interaction energy between NPs and formation fines in porous media. Zeta potential is an input parameter in the calculation of the EDL repulsive force and must be determined experimentally with great accuracy. It is usually measured with a Zetasizer, which uses the electrophoretic mobility (EFM) concept based on the Helmholtz Smoluchowski equation (shown in eq ).

Tables and 8 present the zeta potential data collected from several previously published research articles that show the widespread use of this parameter. It is evident from the data that in all the studies, the zeta potential of the system has been determined with the help of the Zetasizer/zeta potential analyzer, which is a high-technology and expensive apparatus being used worldwide. It measures the electrophoretic mobility and automatically converts it to provide the direct zeta potential value of the system under study. No comprehensive correlation is available to directly calculate zeta potential as a function of system configuration (sand, glass beads, fines, kaolinite, calcite, etc.), salinity, viscosity, pH, and temperature.

Table 7

Zeta Potential Data in the Presence of NPs

									ζ-potential (mV)
authors	year	NP type	NP size (nm)	NP conc.	base liquid	pH	T (°C)	environment	before	after	apparatus/method
Rouxel et al.[132]	2011	Al2O3	13	0.03 wt %	DI water	7.2	25	water	–	34	Zetasizer
Priya et al.[133]	2012	CuO	40–60	0.016 vol %	DI water	12	28–55	water	–	30	Zetasizer
El Badawy et al.[122]	2012	H2–Ag	13	3 vol %	0.03 M NaCl	7		saline	–	–22	Zetasizer
Suganthi and Rajan[134]	2012	ZnO	30–45	1 vol %	DI water	–	25	water	–	49.9	Zetasizer
Habibi et al.[71]	2012	MgO	63	–	DI water	7	40	glass beads	–	–5.68	Zetasizer
			63	0.03 wt %	0.3 M NaCl	7	25	Berea sand		–7	Zetasizer
			63	0.1 wt %	DI water	7	25	glass beads	–34	12.8	Zetasizer
Ahmadi et al.[69]	2013	MgO	63	0.1 wt %	DI water	7	25	glass beads	–44.4	–11	Zetasizer
		SiO2	48	0.1 wt %	DI water	7	25	glass beads		–22.5	Zetasizer
		Al2O3	43	0.1 wt %	DI water	7	25	glass beads		–28.4	Zetasizer
Arab and Pourafshary[65]	2013	ZnO	30	0.03 wt %	DI water	7	25	glass beads	–44	1.57	Zetasizer
		γ-Al2O3	20	0.03 wt %	DI water	7	25	glass beads	–44	0.82	Zetasizer
Arab et al.[67]	2014	MgO	20	0.03 wt %	0.03 M NaCl	6.9	25	glass beads	–27.6	–5.7	Zetasizer
		MgO	20	0.03 wt %	DI water	7	25	glass beads	–44	–1	Zetasizer
Assef et al.[66]	2014	MgO	20	0.0075 wt %	0.02 NaCl	7	25	glass beads	–24.3	6.9	Zetasizer
Bayat et al.[135]	2015	Al2O3	40	0.005 wt %	DI water	6.4	26	water	–	19.1	ζ analyzer
		TiO2	30	0.005 wt %	DI water	6.4	26	water	–	9.1	ζ analyzer
		SiO2	20	0.005 wt %	DI water	6.4	26	water	–	–28.1	ζ analyzer
Alomair et al.[64]	2015	TiO2	50	0.1 wt %	30000 ppm brine	5.9	40	Berea sand	–	–13.2	Zetasizer
		SiO2	15	0.1 wt %	30000 ppm brine	5.9	40	Berea sand	–	–28.3	Zetasizer
		Al2O3	40	0.1 wt %	30000 ppm brine	5.9	40	Berea sand	–	25.3	Zetasizer
		NiO	50	0.1 wt %	30000 ppm brine	5.9	40	Berea sand	–	–23.4	Zetasizer
Sabiha et al.[136]	2016	SWCNT	2500 L	0.1 vol %	DI water	–	25	water		–53.1	Zetasizer
Adil et al.[137]	2016	ZnO	55	0.1 wt %	30000 ppm NaCl	9	95	saline		–20	Zetasizer
Hasannejad et al.[77]	2017	SiO2	145	0.1 wt %	0.3 M NaCl	7	25	glass beads	–28	–25	Zetasizer
Lee et al.[138]	2017	SiO2	37	6.72 wt %	4.5 M NaCl/CaCl2	8	25	saline		–24	Zetasizer
Al-Anssari et al.[117]	2017	SiO2	10	0.1 wt %	1 wt % NaCl	6.3	–	saline	–40	–5	Zetasizer
Abdelfatah et al.[139]	2017	SiO2	10	–	–	6	25	–	–	–22	–
Skoglund et al.[140]	2017	Ag NPs	9	–	DI water	–	25	water	–	–44	Zetasizer
Choudhary et al.[141]	2017	γ-Al2O3	20	0.1 wt %	DI water	7.6	–	water	–	36.7	Zetasizer
Upendar et al.[142]	2018	α-Fe2O3	20	0.05 wt %	0.001 M NaCl	6.5	25	saline	–	9.7	EFM
Kuang et al.[143]	2018	Al2O3	50	0.1 wt %	0.001 M NaCl	7	25	saline	–	39	ζ analyzer
Mansouri et al.[60]	2019	SiO2	15	0.1 wt %	0.03 M NaCl	5.9	25	glass beads	–25	–7.5	Zetasizer
Ma et al.[144]	2019	SiO2-g-SPMA	100	0.5 wt %	5.4 M NaCl + CaCl2	11	170	saline	–	54	Zetasizer
Siddiqui et al.[62,145]	2019	Cu–Al2O3	270	0.01 wt %	DI water	7	23	water	–	48.15	ζ analyzer
Aramendiz and Imqam[146]	2019	SiO2	20	0.75 wt %	DI water	9.5	25	water base mud	–	–34.66	ζ analyzer
Kumar et al.	2020	SiO2–TiO2	15, 20	0.4, 0.05 wt %	DI water	7	90	water	–45.4	–34.1	Zetasizer
Wang et al.[147]	2020	CNCs	70	0.25 wt %	0.175 M NaCl	7	21	saline	–	–60	Zetasizer

Table 8

Zeta Potential Data without NPs

authors	year	base liquid	salinity	pH	T (°C)	environment	ζ (mV)	apparatus/method
Yousef et al.[148]	2012	seawater	600 ppm	7–8	60	carbonate	–13	Zetasizer
Nasralla and Nasr-El-Din[128]	2012	NaCl brine	0.2 wt %	7.7	25	Berea SS	–35	phase-analysis light-scattering PALS
Hussain et al.[36]	2013	NaCl brine	0.5 M	–	25	sand	–20	–
Chen et al.[149]	2014	formation water	14000 ppm	8	25	limestone	–15	phase-analysis light-scattering PALS
Xie et al.[25]	2014	0.2 wt % NaCl	0.2 wt %	8.1	65	sandstone	–23.7	Zetasizer
Xie et al.[118]	2016	NaCl brine	0.2 wt %	–	25	sandstone	–33	–
Yao et al.[150]	2016	distilled water	0	9.2	25	quartz	–23	Zetasizer
Mahani et al.[21]	2017	diluted seawater	0.035 M	7	25	limestone	–11	Zetasizer
Huang et al.[91]	2018	KCl	2 wt %	7	25	coal–brine	–12.61	micro-electrophoresis
Sanaei et al.[94]	2019	NaCl brine	0.01 M	7	25	carbonate	–50	–
Chequer et al.[90]	2019	NaCl brine	0.6 M	7	25	sand	–20	Zetasizer
Takeya et al.[20]	2019	NaCl brine	0.7 M	7.2	50	crude oil	–23	ζ analyzer
Alghamdi et al.[151]	2020	smart water	5761 ppm	7.4	–	carbonate	–8	Smoluchowski equation
Ruan et al.[152]	2020	0.1 M KCl	0.1 M	7	30	clay–brine	4.9	Zetasizer
Takeya et al.[23]	2020	ALSW	0.1 M	7	25	calcite–brine	–3.72	Zetasizer
Tangparitkul et al.[33]	2020	NaCl brine	0.000513 M	–	–	clay–sand	–50	Zetasizer
Gomez-Flores et al.[92]	2020	NaCl brine	0.001 M	7	25	silica–brine	–39	–
Peng et al.[123]	2020	7 mM SDS and 0.5 m NaCl brine	–	–	25	surfactant–water	–80	–

Based on the data available in Table for similar neutral systems (pH ≃ 7–8.5) containing sandstone and NaCl solutions of different Is,[77,90,92,118,128] two correlations are developed to estimate zeta potential for high-salinity and low-salinity conditions, as shown in Figure . Equations and 18 present the developed correlations for low-salinity (0–0.09 M) and high-salinity (0.1–0.9 M approx.) conditions, respectively.

Figure 17

Zeta potential correlations for sandstone.

These correlations are validated for specific salinity range with laboratory data taken from different studies with similar conditions (sandstone, pH ≃ 7–8.5, 25 °C). and the results are shown in Table . Model results are in close agreement with the laboratory measured zeta potential, with less than 10% error for most of the data points, as shown in Figure .

Table 6

Validation of Zeta Potential Correlations

		zeta potential (mV)
data source	Is (M)	lab	model	error (%)
Fogden et al.[129]	0.0171	–39.0	–37.1	4.8
Lebedeva and Fogden[130]	0.075	–36.6	–31.6	13.8
	0.75	–14.7	–14.8	0.7
Hussain et al.[36]	0.05	–30.0	–34.0	13.2
	0.25	–30.0	–27.6	7.9
	0.5	–20.0	–21.2	6.1
Xie et al.[118]	0.1711	–30.0	–29.7	1.1
Shehata Nasr-El-Din[119]	0.8557	–12.0	–12.1	0.8
	0.1	–31.0	–31.2	0.6
	0.0856	–32.0	–30.5	4.6
	0.034	–35.0	–35.5	1.5
Walker and Glover[131]	0.5	–21.0	–21.2	1.1
	0.7	–16.0	–16.1	0.6
Chequer et al.[90]	0.025	–34.0	–36.4	7.0
	0.05	–34.0	–34.0	0.1
	0.2	–24.0	–28.9	20.5
	0.4	–21.0	–23.8	13.3

Figure 18

Close agreement between measured and calculated zeta potentials.

Hence, the generated models can be used to estimate the zeta potential of similar systems, eliminating the need to perform extensive experiments. The calculated zeta potential of sand grains demonstrates the contraction and expansion of the EDLs around fines and sand grains under different salinity conditions. Regarding the practical viewpoint, the developed correlations can considerably benefit the reservoir engineers and production chemists who are involved in the modeling and designing of injection water chemistry/recipe and operation in sandstone reservoirs to avoid fine migration while maintaining designed injectivity.

Conclusions

The DLVO modeling technique based on the quantification and analysis of surface forces is a powerful tool that helps in the analysis of fine migration and control in sandstone reservoirs during low-salinity water injection and alkaline flooding without extensive experimentation. The results can be summarized as follows:

Based on surface force analysis of the FBR system for the solution containing monovalent and divalent ions, this tool can predict the CSC for the injection fluid below which the total interaction energy for the system becomes repulsive and the fine migration starts within the porous media.

Attractive and repulsive forces are affected by fine particle size, ionic strength, types of ions, pH, and the viscosity of the flowing fluid, and all these factors must be considered for the quantification of surface forces.

The application of NPs changes the surface energy and is a promising technique to control and optimize the fine migration with reduced critical salinity. Surface force analysis shows an increase in attractive interaction energy after the application of NPs.

Zeta potential of the FBR system and Hamaker constant are important indicators for any change in attractive and repulsive forces. Precise measurements of these influential parameters can reduce uncertainty in the results and help to provide a better outcome.