In this study, gel flooding process was simulated using Comsol Multiphysics, based on the phase-field method to study the mechanism of hydrogel performance for increasing the oil recovery factor. The developed model was validated with experimental results. Sensitivity analysis was performed on various parameters regarding micromodel and hydrogel, to propose the functionality of the recovery factor with these variables for water flooding and gel flooding processes.
3.1 Validation of Experimental Results for Water Flooding
The modeling result is represented in Figure 3 where oil and water are specified in blue and red, respectively. The results are in good agreement with experimental data from the work of Jamali et al.56. Comparison between recovery factor values for modeling and experimental studies are presented in Table 3. As it is obvious, in layer of low permeability, capillary force is increased and the injected fluid flows to high permeability area. Water injection is continued until the breakthrough time.
Table 3
Comparison of modeling and experimental results for water flooding process
|
High permeable region
|
Low permeable region
|
Total
|
%RF (Experimental)
|
51.46
|
o
|
31.84
|
%RF (Modeling)
|
52.63
|
o
|
32.57
|
%Error
|
2.29
|
o
|
2.29
|
3.2 Sensitivity Analysis on Oil Recovery Factor
A statistical model was developed for prediction of oil recovery factor in water flooding process. In this section, dual effects of a number of parameters were considered.
3.2.1 Statistical Model for Prediction of Recovery Factor in Water Flooding
Sensitivity analysis was performed to determine the effective parameters on oil recovery factor including injection rate, wettability, and viscosities of oil and water. Using these parameters, a statistical model was proposed. The results for ANOVA analysis of variance are reported in Table 4. As it is obvious, P value is less than 0.0001, which denotes the high accuracy of this model. Coefficients of the model are reported in Table 5.
RF= AT + Bq + CƟ + Dµo + Eµw + FTq + GTƟ + HTµo + JTµw + KqƟ+ Lqµo + Mqµw + NƟµo + OƟµw + Pµoµw + Z
|
(15)
|
Table 4
ANOVA analysis of variance to determine oil recovery factor in water flooding
|
Sum of square
|
DOF
|
Mean of Square Errors
|
F Value
|
P Value
|
Model
|
51․95
|
5
|
10․39
|
1020
|
<0․00001
|
Residual
|
26․47
|
26
|
1․02
|
|
|
Table 5
Coefficients of the proposed model using design expert
Coefficient
|
Parameter
|
Coefficient
|
Parameter
|
Coefficient
|
Parameter
|
A
|
0.12
|
G
|
0.016
|
M
|
1.77
|
B
|
3.52
|
H
|
-0.0015
|
N
|
6.7
|
C
|
-10.24
|
J
|
-0.00343
|
O
|
1.32
|
D
|
-9.65
|
K
|
6.6
|
P
|
-0.45
|
E
|
5.34
|
L
|
0.73
|
Z
|
37.7
|
F
|
-0.009687
|
|
|
|
|
In this equation, q represents the flow rate (cc/min), Ɵ is contact angle (degree), µo denotes oil viscosity (cp), µw is water viscosity(cp), and T represents temperature (oC). As it is obvious in Figure 4, the modeling results are in good agreement with the proposed model using design expert, since most of the data are nested near the unit slop line curve.
3.4 Capillary Number
Capillary number (Ca) defined as the ratio of viscous force to surface tension, is one of the dimensionless parameters that is used in flow interpretation. For capillary numbers more than 1, it is increased with increasing the velocity and is negligible for low Reynolds numbers. Capillary number is in the order of 10−6 and is about 1 in the near wellbore region and is defined as follows:
\(Ca= \frac{\mu v}{\gamma \text{c}\text{o}\text{s}\left(Ɵ\right)}\)
|
(16)
|
where \(\gamma\) is the surface tension. Ca is increased with increasing the oil recovery factor. Variations of capillary number with normalized residual saturation is represented in Figure 5 64.
Capillary and viscous forces are in contradiction with each other during oil recovery processes. The capillary force traps the oil in the cavities, while viscous force acts to displace oil by the injected fluid. High sweeping efficiency is an important parameter for successful EOR process which is a function of capillary number. In fact, to improve the recovery factor, capillary number is increased and the mobility ratio is reduced. According to Figure 5, for low capillary numbers, the diagram is linear with zero slope. In critical point which is a function of wettability, permeability and pore structure, a sharp reduction is observed. Accurate representation of capillary number functionality with normalized saturation is an essential tool for simulation of oil sweeping and minimizing the residual oil saturation. Capillary number diagram in terms of normalized saturation is mostly related to reservoir heterogeneity, but it is recommended to use its model for investigation on sweeping process. Figure 6 illustrates the capillary number diagram in terms of normalized saturation for the modeling results in the present study. As it is obvious, capillary number pattern is consistent with the one presented in the literature.
3.5 Hydrogel Flooding Simulation
As represented in previous sections, after water breakthrough, the prepared hydrogels are injected to micromodel. By injection of hydrogels, the high permeability region is blocked and water is transferred to non-swept areas of low permeability. Figure 7 illustrates the modeling result using Comsol Multiphysics software which are in good agreement with experimental study.
In simulation studies, density of hydrogels was equal to that of displacing fluid and its viscosity was 30cp. During injection of hydrogels, they might break or transform. Table 6 represents the RF% for experimental and modeling studies. Thickness of the contact surface in simulation studies was obtained 111.7 micrometers. The phase transform parameter in Cahn-Hillard equation (Mo) was equal to 5*10-13 m3/s. The excess free energy (\(\wedge\)) was reported -0.0003 J/m3, and the interfacial tension was 2×10-5Pa.s. This table indicates that simulation and experimental results are in good accordance with each other. Figure 8 illustrates the variations of oil recovery factor for the whole process.
Table 6
Comparison of the results for experimental and simulation studies in gel flooding process
RF
|
High-Perm region
|
Low-perm region
|
Total
|
Experimental
|
87․54
|
13․79
|
57․05
|
Simulation
|
84․02
|
13․22
|
53․99
|
Error (%)
|
4․02
|
4․13
|
4․06
|
3.5.1 Mechanisms for Hydrogel Movement in Porous Media
The mechanisms of hydrogel movement in porous media include breaking, deformation, and blocking, which are illustrated in Figures 9a to 9c, respectively. All of these mechanisms are effective on the performance of the injected hydrogels, which are reviewed in this section.
Hydrogel breakage is mostly occurred within the throats near injection point. The broken hydrogels are then penetrate more deeply in porous media. Hydrogels are broken by increasing the pressure to more than a threshold value, otherwise hydrogels would block the throats. Pressure variations at two sides of micromodel is represented in Figure 10. Two regions are defined in this figure. In the first zone, hydrogel particle blocks the throat, temporarily. The pressure is increased at inlet and reduced at outlet. This leads to increasing the pressure difference in micromodel. As a result, hydrogel particle is deformed and broken (Figure 9a-part b). The two generated particles are get to the next pore (Figure 9a-part c) and one of the particles enter to the next throat (Figure 9a-part d). As a result, the outlet pressure is slightly increased (Zone II). In all the steps for breakage and deformation, gels would plug the throat, if the pressure is not increased enough.
Hydrogels might also cross the throats by deformation, which actually happen at injection point and in medium depths of the reservoir. Figure 11 shows pressure variations at two sides of micromodel for deformation of hydrogels. Two regions are defined in this figure. In the first region, hydrogels are blocked the throat. This causes increasing the pressure difference in this region. This pressure difference leads to deformation and movement of hydrogel particles in the throat. The movement of hydrogels is continued toward the next pore and throats (Figure 9b-part d). This is obvious with an incremental increase in outlet pressure (region II).
While the pressure variations is not enough for deformation or breakage of hydrogels, the blockage is the predominant mechanism. Blockage is mostly occurred at depth of the porous media by reducing the pressure gradient. Figure 12 shows pressure variations at two sides of micromodel for the mechanism of blockage. Hydrogel deformation is occurred at the first region by pressure variations (region I). For pressures less than the threshold value, blockage is occurred at the throat and pressure value at two sides of micromodel remains almost constant (region II). For the first region of Figure 12, the pressure difference between two sides of micromodel is increased since hydrogels block the throat. The pressure increase leads to deformation of hydrogel, but the pressure is not high enough for breaking the hydrogels or passing them through the throats. Since the throat is blocked, the inlet and outlet pressure values, remain constant which is obvious in region II.
In this section, pressure difference between the two ends of micromodel is discussed for hydrogel flooding process. The first section of Figure 13 represents the pressure variations in micromodel for water flooding at various time steps. The pressure drop is reduced during water flooding. The next part represents the pressure variations for hydrogel flooding. After water breakthrough, during hydrogel injection, the pressure difference between inlet and outlet, represents considerable increase. Blockage of some of the throats during hydrogel injection causes this phenomenon. Moreover, there is distinctive pressure fluctuations after hydrogel injection. This is due to permanent or temporary throat blockage during hydrogels injection.
3.5.2 Pressure Profile Diagram
To obtain pressure profile along the micromodel, the pressure variations along line A is investigated (Figure 14). The pressure changes during water flooding is represented in Figure 15 at various time intervals. As shown in the figure, when the initial parts of micromodel are swept, the pressure variation along the micromodel is decreased at various time steps and water is replaced by oil. At 1320 seconds, the pressure profile with almost monotonous slope is obtained. This means that breakthrough has occurred.
3.5.3 Sensitivity Analysis on Oil Recovery Factor for Hydrogel injection
In this section, effective parameters on oil recovery factor in hydrogels injection process are analyzed. These parameters include mean diameter of hydrogel particles, hydrogel injection rate, and hydrogel viscosity.
3.5.3.1 Hydrogels Mean Radius
Hydrogel size is an important parameter in determining the penetration depth and hydrogel performance in increasing the oil recovery factor. The final radius of the injected hydrogel is a function of its initial size and the swelling ratio. Depending on the method of hydrogel synthesis and its chemical compounds, the blocking ability is different. Bai et al. 16stated that particles four times the throat diameter, are able to pass the throat. Larger particles should be broken before passing the throat. However, the operating conditions and hydrogel chemistry are of important parameters. Depth of penetration is a function of particle size, since it is effective on injectability of the fluid.
In the present study, effect of hydrogel radius on oil recovery factor was studied. The mean radius of hydrogel particles in experimental study was 69.9 µm and the average size of throats in high and low permeability areas were 350 and 240 µm, respectively. Sensitivity analysis are performed at constant injection rate of 1e-3 cc/min and viscosity of 30cp. The results are represented in Figure 16. Increasing the size of hydrogels led to increasing the percentage of blocked throats in high permeability region, which results in alteration the direction of the injected fluid toward the low permeability region. Therefore, the oil recovery factor would increase in this section and the amount of produced water is controlled. Of course, the size of hydrogels should not be so large that could not enter the pores. By increasing the hydrogels radii, their ability to penetrate to micromodel depth is decreased, and blockage is occurred in areas close to the injection point. This would reduce the sweeping efficiency in high-permeability region. Variations of the recovery factor in the whole micromodel, is also represented in this figure.
3.5.3.3 Hydrogel Injection Rate
The functionality of recovery factor with injection rate is analyzed from different standpoints. The first one is the effect of injection rate on functionality of hydrogel particles, and the other is the effect of injection rate on displacement process similar to that of two-phase flow. Result of the two parameters is represented in Figure 17. In this study, the mean radius of hydrogels was 69.9µm and the solution viscosity was 30cp. As represented in this figure, the recovery factor of the high permeability region is increased with increasing the injection rate up to an optimum value (1×10−3 cc/min). The recovery factor in low permeability region is at its minimum value at this point. The total oil recovery factor is increased with increasing the injection rate.
3.5.3.4 Hydrogel Viscosity
Hydrogels are considered as a high viscosity liquid for simulation studies. To simulate solid particles, Mirzaei 65 estimated their viscosity to be about 100 times that of a liquid. In Saghafi et al. 66 study, the viscosity of hydrogels for simulation studies was 80 cp. The flow direction could be changed due to high viscosity of the injected hydrogels. In the present study, viscosity of the injected hydrogels was 30cp. This value leads to an acceptable penetration depth for hydrogel particles to change the flow direction and increasing the oil recovery factor. Effect of hydrogel particles viscosity was studied on oil recovery factor in high and low permeability regions and the whole micromodel. In this study, the injection rate and hydrogel radius were 1e-3 cc/min and 69.9 µm, respectively. Effect of this parameter on oil recovery factor, is illustrated in Figure 18. With increasing the viscosity of hydrogels, the flow in high permeability region is stopped near the injection point and leads to deviation of the flow to low permeability region. This leads to reducing the oil recovery factor in high permeability region, since hydrogels could not penetrate to the acceptable depth in micromodel. The flow resistance is increased in high permeability region and the flow is directed to low permeability part which is represented in Figure 19 (path 1). Moreover, hydrogels are accumulated at inlet of high permeability region which steers the flow to low permeability area (path 2). With increasing the viscosity of hydrogel particles, the penetration and placement of the particles in high permeability region is reduced. This leads to reducing the oil recovery factor, since hydrogels are unable to direct the flow to low permeability area.
3.5.4 Statistical Model for Hydrogel Injection
In this section, a statistical model is represented to determine oil recovery factor in hydrogel injection process. This model was proposed based on the effective parameters including, fluid injection rate, hydrogel size and viscosity of hydrogel solution. The results of ANOVA analysis for the high permeability region is represented in Table 7. F-values and P-values represent that the model is significant. The proposed model is represented in Eq. 17.
RF−3=-0.000133Q + 1.11783E-09r + 1.67641E-09µ + 2.37196E-06Qµ + 1.81184E-06
|
(17)
|
Table 7
ANOVA Analysis for the recovery factor in high permeability region
|
|
Sum of squares
|
DOF
|
Avg of Sum of squares
|
F value
|
P value
|
High-perm region
|
Model
|
3.380E-14
|
4
|
8.450E-15
|
55.80
|
0.0038
|
residual
|
1.514E-16
|
3
|
4.543E-16
|
|
|
where Q represents the injection rate of hydrogels (cc/min), r denotes the hydrogel radius (µm), and µ shows the viscosity of hydrogels (cp). The fitting statistics are reported in Table 8.
Table 8
Fit statistics for the proposed model for high permeability region.
Std. Dev.
|
1.231E-08
|
R²
|
0.9867
|
Mean
|
1.890E-06
|
Adjusted R²
|
0.9691
|
C.V. %
|
0.6510
|
Predicted R²
|
0.9057
|
|
|
Adeq Precision
|
20.5770
|
The same procedure was applied for the low permeability region and oil recovery factor was proposed in terms of injection rate, viscosity, and hydrogel radius by the following equation. The fitting statistics are also reported in Table 9. The results of ANOVA analysis for the low permeability region is represented in Table 10.
RF−1.5=2.64543Q - 0.000010r -0.000183µ + 0.024399Qr+0.020666
|
(18)
|
Table 9
Fit statistics for the proposed model for low permeability region.
Std. Dev.
|
0.0001
|
|
R²
|
0.9995
|
Mean
|
0.0148
|
|
Adjusted R²
|
0.9988
|
C.V. %
|
0.6081
|
|
Predicted R²
|
0.9963
|
|
|
|
Adeq Precision
|
92.8034
|
Table 10
ANOVA Analysis for the recovery factor in low permeability region
Source
|
Sum of Squares
|
DOF
|
Average Sum of Squares
|
F-value
|
p-value
|
Model
|
0.0000
|
4
|
0.0000
|
1432.06
|
< 0.0001
|
Residual
|
2.436E-08
|
3
|
8.121E-09
|
|
|
Figures 20a and 20b, represent the results of modeling and prediction of oil recovery factor based on the proposed statistical models for high and low permeability regions, respectively. As represented in these figures, most of the data are nested near the unit slope line curve which states the high accuracy of the proposed statistical model.