Investigation of Nano-Silica Solution Flow through Cement Cracks

Abstract

Cement cracks are one of the most common failures in oil and gas wells. Cracks can reduce cement strength, resulting in a loss of zonal isolation and fluid leak. Placement of gels of nanoparticles (NPs) in the cracks is considered as a promising solution to solve the problem. It is highly desirable to know if the flow behavior of the NPs solutions is predictable when they are squeezed into the cracks. Experimental tests were performed in this study to investigate the flow behavior of nano-silica solutions in ducts of cross-sections of rectangular shape. The linear relationship between flow rate and pressure gradient and the calculated Reynolds number values suggests laminar flow in the ducts. However, the Hagen–Poiseuille correlation for laminar flow does not describe the flow behavior of the nano-silica solution. The classic hydraulic model with hydraulic diameter describes the nano-silica flow behavior with an average error of 12.38%. The cause of discrepancies between the flow models and the measured data is not known. It can be attributed to the NPs–NPs frictions and NPs–wall frictions in the rough ducts that were not considered in the flow models.

1. Introduction

Nanoparticles (NPs) have recently been used in the oil and gas industry in the form of NP solutions. Mohamadian et al. [1] published their experimental study on rheological and filtration characteristics of drilling fluids enhanced by NPs with selected additives. Gu et al. [2] reported shear thickening effects of drag-reducing NPs for low permeability reservoir. Khalil et al. [3] presented a review of design and challenges in NPs applications. Babakhani et al. [4] presented a review of continuum-based models and concepts for the transport of NPs in saturated porous media. Yekeen et al. [5] presented a comprehensive review of NPs applications for hydraulic fracturing of unconventional reservoirs. NPs have recently been considered for sealing flow channels and cracks in wellbore cement sheath [6,7,8].

Silica NPs also have been investigated as an additive for polymer gel. The applications can be found in wellbore strengthening and wellbore remediation [9]. Adibnia and Hill [10] addressed that the cross-linking level of polymer gel would increase in the presence of silica NPs. The hydrogen bonding between silica NPs and polymer gel enhances the polymer gel’s rheology to make the gel become more solid-like [11]. Liu et al. [12] concluded that the addition of silica NPs would enhance gel strength and gel thermal stability. The improved gel strength can be illustrated by accumulations and arrangements of NPs in the polymer chain bunches and gel structure meshes during the cross-linking stage. In comparison, the enhanced gel thermal stability is attributed to the considerable number of hydroxyl groups (OH⁻) on the silica NPs surface and the negative charges of the silica NPs in the gel solution.

Because of the narrow nature of the channels and cracks, there is concern about the placement of the NPs due to the flow resistance. In other words, the transport of NPs to the downhole becomes one of the most concerning issues in sealing cement cracks. Many groups have studied the transport of NPs in porous media [13,14,15]. Song et al. [16] investigated the effect of NPs size on NPs transport in porous media, while Ma et al. [17] studied the impact of NPs shape. The tumbling motion of NPs rod-shape transforms their translational and rotational trajectories. Tang and Cheng [18] analyzed that the electrostatic attraction between NPs and porous media would intervene in NPs mobility.

The channels and cracks in well cement sheaths have a cross-sectional area of slot-like shapes. Guo et al. [7] characterized these cross-sections into two categories: rectangular shape and bow shape. The bow-shaped cross-section was considered for calculations of cross-sectional area and the average velocity only. A general equation for pressure-drop calculations was used based on the average velocity of fluid. Spiga and Morini [19] presented a symmetric solution for velocity profile in laminar flow through ducts of rectangular cross-sections for Newtonian fluids. The solution describes two-dimensional velocity distribution using the finite Fourier transform. They claimed that the solution provides more accurate numerical results in comparison to the well-known velocity distributions usually quoted in the literature. Because Spiga and Morini’s [19] solution takes a very complex form, it is not widely used in engineering analysis. Guo et al. [7] published an approximate solution for fluid flow assuming two-dimensional velocity distribution in cracks of finite heights. It is expected to give improved accuracy compared to the classic solution [20] derived from one-dimensional velocity profile for fluid flow between two parallel plates based on Hagen–Poiseuille’s principle for laminar flow.

Laboratory testing was conducted in this study to investigate the flow of nano-silica solutions in ducts of rectangular cross-section. It was found that the result does not follow the flow equations of Jousten [20] and Guo et al. [7]. The models under-predict the pressure drop by over 500%. The test data can be described by the classic hydraulics equation [21] when the concept of hydraulic diameter is employed. However, the classic discrepancy is attributed to the NPs-NPs frictions and NPs-wall frictions and the effect of roughness of the sandstone surface that was not considered in the equations of Jousten [20] and Guo et al. [7].

2. Experimental Investigation

2.1. Experimental Apparatus

Figure 1 illustrates the schematic of experimental setup used in this study. The central component of the experiment system is the core holder (b) shown in Figure 2 that holds a “cracked” cement core sample with confining pressure.

The laboratory experimental setup is presented in Figure 2. The Nitrogen tank (a) will supply the confining pressure of 400 psi to the core holder (b) where the cracked cement core is placed. Then, the Isco Pump (e) will push the nano-silica solution from the accumulator (c) through the cracked cement core, and the pressure drop is measured by the pressure transducer (4) every minute and recorded in the computer system (f).

2.2. “Crack” Preparation

A 22-inch long and 2-inch diameter core was cut along its diameter and split. Metal was then placed along the edges (Figure 3) to separate/support the two pieces. The metal plate is 0.5 inches wide and 0.0625 inches thick. The two pieces of core halves were then wrapped with tape (Figure 4). The non-supported area of the pieces forms a cap, simulating a crack.

3. Experimental Procedure

The experimental procedure is outlined as follows:

• Prepare the core: The core is cut to 22 inches long and 2 inches in diameter.
• Prepare the crack: The cut core halves are separated by 2 pieces of metal plates, forming a 0.0625 inches gap to simulate a crack.
• Prepare the nano-silica solution: The 2.5%, 3.75%, and 5% w/v nano-silica were prepared by diluting the 50% w/v original nano-silica solution (Figure 5). The viscosities of the solutions are 1.2 cp, 1.3 cp, and 1.4 cp, respectively. The nano-silica particle size is 50 nm and Nouryon Pulp and Performance Chemicals Inc provided the original solution.

• Place the core and nano-silica solution into the system: Install the core in the core holder and pump the nano-silica solution into the accumulator.
• Inject the nano-silica solution through the “crack”: After applying the confining pressure of 400 psi, the nano-silica solution is pumped through the “crack” at designed flow rates.
• Record pressure drop: Pressure drop data are measured every minute during fluid injection.

4. Experimental Result

To investigate the nano-silica fluid flow in cement cracks, injectivity tests were carried out with various nano-silica solution volumetric flow rates and nano-silica solution concentrations to examine the influencing parameters to induce a pressure drop. The experiments were conducted under the confining pressure of 400 psi. The tested volumetric flow rates are 5 mL/min, 7.5 mL/min, 10 mL/min, and 12.5 mL/min, and the nano-silica solution concentrations, C, are 2.5%, 3.75%, and 5%. The pressure drop was observed every minute. The test result is summarized in

Table 1. Result of nanoparticle solution flow tests.

Concentration of Nano-Silica Solution (%)	Pressure Gradient (psi/ft)
	q = 5 mL/min	q = 7.5 mL/min	q = 10 mL/min	q = 12.5 mL/min
2.5	0.65	0.98	1.42	1.75
3.75	0.76	1.10	1.47	1.8
5	0.82	1.15	1.53	1.85

. All pressure drop profiles of nano-silica solution flow are presented in Figure 6, Figure 7 and Figure 8. In general, the increase in volumetric flow rate would lead to an increase in pressure gradient following the Hagen–Poiseuille correlation for laminar flow of Newtonian fluid flow in the tubes. The calculated Reynolds number between 4.7 and 11.8 confirms laminar flow. For example, at the nano-silica solution concentration of 2.5% and the volumetric flow rate of 5 mL/min, the pressure drop through the “crack” developed over time and stabilized at 0.65 psi. With the increase in volumetric flow rate to 7.5 mL/min, 10 mL/min, and 12.5 mL/min, the stabilized pressure drop surged to the new values of 0.98 psi, 1.42 psi, and 1.75 psi, respectively. On the other hand, it seems likely that the nano-silica concentration did not contribute significantly to the pressure drop. In fact, the pressure drop did not fluctuate much, corresponding to the change in nano-silica concentration. A plot of pressure gradient versus flow rate is shown in Figure 9. The linear trend of data in the figure indicates a proportionality between pressure drop and flow rate, which is a behavior of the laminar flow.

Figure 10 plots the pressure gradient data versus nano-silica concentration at different flow rates. It indicates that the pressure gradient is not sensitive to the nano-silica concentration in the range from 2.5% to 5%. The sensitivity is slightly higher in the low-flow rate end than that in the high-flow rate end.

5. Discussion

Guo et al. [7] presented the following mathematical model for nano-particle transport through cracks based on Hagen–Poiseuille’s principle for laminar flow:

$$\frac{dp}{dz}=\frac{12\mu v_{av}}{w^2}\left(\frac{h+w}{h}\right)$$

(1)

where p is the pressure, z is the crack length, µ is the fluid viscosity, h is the crack height, and v_av is the mean flow velocity defined by:

$$v_{av}=\frac{Q}{A}=\frac{Q}{wh}$$

(2)

where Q is the volumetric flow rate of fluid, A is the cross-sectional area of the crack, and w is the average width of crack of rectangular cross section. The variables in Equations (1) and (2) are in consistent units. When the pressure gradient data derived from our tests were compared with that predicted by the model, a great discrepancy was found. The test-derived pressure gradients are over 500% higher than the model-predicted pressure gradients. The reason for the discrepancy was first attributed to the effect of roughness of the sandstone surface that was not considered in the derivation of Guo et al.’s [7] model.

The test result was also compared with the hydraulics equation for laminar flow where the Fanning friction factor is expressed as an explicit function of Reynolds number (f_F = 16/N_Re). The hydraulics equation takes the following form after arrangement using the expression for Reynolds number [21]:

$$\frac{dp}{dz}=\frac{\mu v_{av}}{D_h{}^2}$$

(3)

where p is the pressure in lbf/ft², z is the crack length in ft, µ is the fluid viscosity in cp, v_av is the mean flow velocity in ft/s, and D_h is the hydraulic diameter in ft defined by:

$$D_h=\frac{4 \times Cross Sectional Area of Flow}{Wetted Perimeter}$$

(4)

For a crack with height h and width w, the D_h is expressed as:

$$D_h=\frac{2hw}{h+w}$$

(5)

Table 2. Comparison of observed pressure gradients and that given by Equation (3).

2.5% Nano-Silica Solution
Flow Rate	Velocity	Pressure Gradient (psi/ft)		Difference (%)
(mL/min)	(ft/s)	Test	Equation (3)
5	0.007	0.65	0.59	−9.68
7.5	0.010	0.98	0.88	−10.14
10	0.014	1.42	1.17.	−17.31
12.5	0.017	1.75	1.47	−16.13
3.75% Nano-Silica Solution
Flow Rate	Velocity	Pressure Gradient (psi/ft)		Difference (%)
(mL/min)	(ft/s)	Test	Equation (3)
5	0.007	0.76	0.64	−16.31
7.5	0.010	1.10	0.95	−13.27
10	0.014	1.47	1.27	−13.46
12.5	0.017	1.80	1.59	−11.66
5% Nano-Silica Solution
Flow Rate	Velocity	Pressure Gradient (psi/ft)		Difference (%)
(mL/min)	(ft/s)	Test	Equation (3)
5	0.007	0.82	0.68	−16.47
7.5	0.010	1.15	1.03	−10.66
10	0.014	1.53	1.37	−10.46
12.5	0.017	1.85	1.71	−7.44

shows the measured and model-calculated data for the nano-silica solutions. It indicates that Equation (3) under-predicts pressure gradient by 7.44% to 17.31%, averaging 12.38%. The cause of discrepancy is attributed to the NPs–NPs frictions, NPs–wall frictions, and the effect of roughness of the sandstone surface that was not considered in the equation.

6. Conclusions

Experimental tests were performed in this study to investigate the flow behavior of nano-silica solutions in ducts of cross-sections of rectangular shape. The following conclusions are drawn.

The linear relationship between flow rate and pressure gradient and the calculated Reynolds number values suggests laminar flow in the ducts. However, the Hagen–Poiseuille correlation for laminar flow (Equation (1)) does not describe the flow behavior of the nano-silica solution.

The classic hydraulic model with hydraulic diameter (Equation (3)) describes the nano-silica flow behavior with an error between 7.44% and 17.31%, averaging 12.38%.

The cause of discrepancies between the flow models and the measured data is not known. It can be attributed to the NPs–NPs frictions and NPs–wall frictions in the rough ducts that were not considered in the flow models.