Evaluation of Na-silicate adsorption on surface minerals helps to determine the fraction of the injected Na-silicate that reaches the targeted treatment location in the reservoir. Dynamic adsorption was done by injecting a Na-silicate slug with the size of 0.48 PV in sand-pack A (Table 3), which is initially saturated with DW. The slug size was almost similar to what is later used for gel treatments in dual-permeability sand-packs. Two different Na-silicate solutions were studied which had Si concentrations of 33 mg/L and 3.9 mg/L, which correspond to 0.0093 wt% and 0.0011 wt% Na-silicate solutions, respectively. The low Si concentration in the tested solutions was used to precisely estimate the adsorbed amount. The concentration of Si in the effluent was measured using ICP, and the adsorption measurements were performed at room temperature.

The ratios between Si concentrations in the effluent and the injected concentrations (C/C₀) are plotted against the injected PV (Figure 6a,b for the two concentrations of 33 mg/L and 3.9 mg/L, respectively). Numerical simulation was used to match the experimental data. The matching parameter was adsorption density, as shown in Figure 6. The adsorption density was increased from 0 (for non-adsorbing material such as Cl⁻ tracer) to higher values, to match the experimentally obtained concentration curves. The simulation method addresses only the adsorption process and not the desorption process. In both cases, the experimental data were matched numerically only when it was assumed that Si adsorption takes place at (C/C₀) higher than 0.25.

The adsorption density was obtained from the numerical matched experimental data as shown in Figure 6. Figure 6a demonstrates that for C₀ = 33 mg/L, the best numerical match occurred when adsorption density is about 8.5 × 10⁻¹⁰ kg/cm³. In the case of C₀ = 3.9 mg/L (Figure 6b), the obtained adsorption density was about 4.5 × 10⁻¹⁰ kg/cm³. As Si concentration in the injected slug increases from 3.9 mg/L to 33 mg/L (almost 8 times higher), the adsorption density is almost doubled.

Mono-layer adsorption density of Si in sand pack A (with porosity of 0.45) was obtained as 1.4 × 10⁻⁸ kg/cm³ by a simplified estimation. To estimate mono-layer adsorption, it was assumed that the average grain diameter in the sand-pack A is 200 μm. To simplify the calculations, the sand grains were considered to have spherical shape. It was also assumed that Si is adsorbed in the form of silicate molecules (SiO₂) with atom radii of 0.16 nm, and the adsorption takes place on the total surface (including the contact area) of all the grains in the sand-pack. The experimentally obtained Si adsorption densities (Figure 6a,b) are more than ten times smaller than the estimated monolayer value. This demonstrates an inconsiderable Si adsorption on sand grain surfaces, especially at high Si concentrations, such as Na-silicate solutions which are used for gel treatments (5 wt% Na-silicate solution has Si concentration of 17,000 mg/L).

Figure 6 shows that Si adsorption is reversible. This is implied from the tails of the concentration profiles, which starts at (C/C₀) of about 0.25. It means that the adsorbed Si is desorbed in the post-flush water. The mass balance between the adsorbed and desorbed Si showed that most of the adsorbed mass is desorbed in the following water. For example, in the case of C₀ = 3.9 mg/L, the total adsorbed Si amount in the sand-pack is about 0.015 mg. If it is assumed that the desorption starts at 1.5 PV with an average concentration ratio of C/C₀ = 0.1 (as shown in Figure 6b), after about 2 PV, approximately 0.0065 mg (almost half) of the adsorbed mass is desorbed.

Na-silicate is water soluble; hence it diffuses into the water during the transportation in the porous media. Furthermore, since there is density and viscosity contrasts between Na-silicate solution and fluid (water) in the porous media, the Na-silicate slug propagation is not piston like and the shape of the formed gel is not regular. A numerical sensitivity study was performed on the effects of physical diffusion, density and viscosity contrasts on the slug propagation in porous media. Figure 7 compares the relative concentration (C/C₀) profiles of the injected solution slug at two different times and two different heights in the simulated medium, i.e., at the bottom (solid lines) and the top (dashed lines). The results are shown for two different solutions. First one (blue lines) only has density contrast with the water in the medium (density ratio = 1.2), while the other one (red lines) has both density and visosity contrast with the water in the medium (density ratio = 1.2 and viscosity ratio = 1.3). As shown in Figure 7, the injected slugs are diffused in water as they transport in the medium. The effect of numerical diffusion was minimized by small grid sizes. The effect of diffusion has to be considered in the design of in-depth treatments.

For a solution with density contrast of 1, the concentration profiles at the bottom and top of the medium coincide. However, when the solution density ratio increases to 1.2, gravity segregation affects the concentration profile as the slug propagates inside the porous medium (Figure 7, blue lines). The relative concentration profile is thicker at the bottom and thinner at the top. It causes the post-flush water to override the slug at later times. The effects of gravity segregation and water override are more pronounced as the height of the medium increases.

Figure 7 shows that the concentration profiles are also affected by viscosity contrast between the injected solution and the formation water (Red lines). It is observed that the viscosity contrast exacerbates the gravity segregation effect which increases the intensity of the water override. When the slug has a higher viscosity than the formation water, the flow resistance is higher wherever the slug is thicker. Therefore, since the slug is thinner at the top, due to the density contrast, flow resistance is lower there, compared to the bottom of the medium. Hence the flow velocity is faster at the top and water override is accelerated.

The Na-silicate gel profiles and strengths for different injected slug sizes are studied. For all the experimental studies in the coming sections, 5 wt% Na-silicate solutions with different pH values were used. Na-silicate samples with lower contents than 5 wt% did not produce strong enough gels for in-depth applications.

5 wt% Na-silicate at pH = 10.40 were injected with different slug sizes, targeting the middle of the sand-pack A. Sand-pack was initially saturated with DW. The Na-silicate slug was injected at a rate of 0.1 mL/min, which is corresponding to the shear rate of about 10 s⁻¹. Silicate solution was placed almost at the middle of sand-pack with a DW post-flush. The injection rate was then reduced to 0.01 mL/min to monitor the gel formation by measuring the pressure drop between the inlet and the outlet. The results are compared for four different slug sizes of 0.35, 0.4, 0.45 and 0.5 PV. Figure 8 shows the numerically and experimentally obtained profiles of the gel for different Na-silicate slug sizes after the gel placement at the middle of the medium. Figure 8a is a picture taken from the formed gel in the used transparent tubes after removing the sand in other parts using high pressure air. There is a good agreement between the experimental (Figure 8a) and simulated (Figure 8b) results, which shows the ability of the developed numerical method in predicting the gel propagation and profile. The profile of the formed gel is dependent on the Na-silicate slug size. In the case of 0.35 PV slug size, most of the injected Na-silicate was diluted, mostly at the top due to gravity effect; so the formed gel has a small size. As the injected slug size increases, the formed gel sizes increased. For slug sizes ≤ 0.4 PV, water override was clear, while for the larger slug sizes, the gel fully occupies the sand-pack cross section.

Figure 9 compares the pressure build-up (flow resistance) during gel placements with different slug sizes. The bulk measured viscosity profile of 5 wt% Na-silicate at pH = 10.40 is also presented in Figure 9. No considerable pressure increase is observed prior to Na-silicate gelation As shown in Figure 9, the pressure build-up in the sand-pack for Na-silicate slug sizes of 0.45 and 0.5 happens at about t = 200 min, in a good agreement with bulk measured gel time (t_g = 180 min). It takes slightly more time after gelation in sand-pack to be sensed by the pressure gauge at low injection rates. As depicted in Figure 9, Na-silicate slug sizes ≤ 0.4 cannot increase the flow resistance in the sand-pack after the gel placement, because of high water override (Figure 8). In the case of larger slugs of 0.45 PV and 0.5 PV, pressure increases after the gel setting time and then stabilizes at different values, as shown in Figure 9. The pressure stabilization indicates that a water communication path is formed between the two sides of the gel at about t = 800 min, which cannot be detected in Figure 8. By increasing pressure behind the Na-silicate slug, water gradually overcomes the viscous force of the newly formed gel at the top.

As shown in Figure 10, water override happened by displacement of small volume of water at the top of the formed gel through a thin layer. Residual resistivity factor (F_rr) for 0.35 PV, 0.4 PV, 0.45 PV and 0.5 PV slug sizes is 1, 1.7, 18.5 and 650, respectively. F_rr was determined by the relative pressure drop in the sand-pack after gel placement and before gel treatment.

First, a sensitivity study was done on the size of LSW pre-flush in sand-pack A, using numerical simulation. The model was initially saturated with SSW, then a LSW pre-flush was injected followed by 0.48 PV Na-silicate slug with the injection rate of 0.1 mL/min. Figure 11 shows relative water salinity and Na-silicate relative concentration as function of the sand-pack PV for three different LSW pre-flush sizes. The results are shown at the end of Na-silicate slug injection. LSW, which is 25 time diluted SSW, has relative salinity of 0.04. It can be seen that 0.25 PV pre-flush did not make a proper space between SSW and the injected Na-silicate slug, hence there is a possibility of plugging due to precipitation. 0.5 PV LSW pre-flush, however, is adequate to make a safe space between the silicate slug and the brine within the tested sand-pack.

Dual permeability sand-pack flooding experiment was done both experimentally and numerically. The medium (including both sand-packs A and B) was initially saturated with red dyed oil (n-C₁₀ and 0.01 M NN-DMDA) and colorless irreducible SSW. Primary water flooding was done by injecting SSW for 0.5 PV (of both sand-packs) with flow rate of 0.1 mL/min. The measured and simulated pressure drops along the parallel sand-pack system are depicted in Figure 12. At the beginning of water flooding, pressure drop gradually increased to about 1.5 kPa, and then stabilized. The pressure drop was almost constant (around 1.5 kPa) during the whole water injection process, as shown in Figure 12.

Water preferentially swept the oil in the high permeability sand-pack, while low permeability sand-pack remained unswept, except small parts close to the inlet. As shown in Figure 13a, the simulated pressure distributions in both sand-packs are similar during water-flooding period. However, since a given pressure drop results in a higher flow rate in high permeability sand-pack, the major oil production occurs in this sand-pack.

After water breakthrough in sand-pack B, the oil production became negligible, while water production dominated, as shown in Figure 14. Water flooding oil recovery factor was less than 50%.

Na-silicate gel was used to modify the flow profile and to recover the remaining oil in low permeability sand-pack. LSW pre-flush was injected for 0.25 PV (of both sand-packs) to dilute the ions of SSW in the medium. It was then followed by injecting 0.4 PV of 5 wt% Na-silicate solution with pH = 10.70. According to the prediction method presented in our previous study [16], the gel time for this sample (5 wt% at pH = 10.70) at room temperature and zero salinity is t_g0 ≈ 30 h. The is high enough to ensure that the gelation of Na-silicate sample would be delayed to place the gel at the middle of sand-pack, while it mixes with LSW and becomes hot under T = 50 °C. The gelation time of the injected sample decreases to about 0.2 when t_g0 =6 h at T = 50 °C and in contact with LSW, which is suitable for gel placement at the middle of sand-pack.

As shown in Figure 12, the measured pressure drop during Na-silicate injection was about 1.7 kPa (approximately 13% higher than that of water flooding), indicating good injectivity of Na-silicate prior to gelation. It was also observed that the injected Na-silicate only flows through the high permeability sand-pack, except small amount which penetrates in low permeability sand-pack, mostly due to diffusion. LSW was injected afterward to displace the Na-silicate from the inlet and also to dilute the Na-silicate leakoff into the low permeability sand-pack. Then injection rate was reduced to 0.001 mL/min to set the gel.

Water injection rate was increased to 0.1 mL/min at t = 1200 min. Upon increasing water injection flow rate, pressure increased sharply to a higher level (Figure 12) and the injected water flowed through the low permeability sand-pack (A) and swept the remaining oil, as shown in Figure 14. Figure 13b shows that the pressure drop across both sand-packs remained similar; however with different pressure distributions in porous media. There is a high pressure behind the formed gel in sand-pack B, while there is a pressure gradient in sand-pack A, due to water flow with constant rate. Na-silicate gel treatment enhanced oil recovery factor to more than 80%.

Figure 15 shows the Na-silicate slug size and post-flush water amount to make a 180 ft gel at the middle of the HP layer of the studied reservoir.

The results are compared for five different reservoir models, given in Table 5. Case (a) is considered as a reference case. The Na-silicate solution was injected after 1 PV water flooding. Figure 15 reports the slug sizes as PV of the HP layer. The first columns in Figure 15 demonstrate the calculated Na-silicate and post-flush water slug sizes in the case without dilution or leak off into the LP layer.

Figure 16 illustrates the concentration ratio of the injected Na-silicate for different studied cases at three moments: the end of Na-silicate slug injection, the meanwhile to the target, and at the targeted position. It is supposed that the gel is formed only when the normalized concentration of Na-silicate is higher than 0.7. This is a realistic assumption, since a practical Na-silicate content for field applications is about 5 wt% and Na-silicate never gels if it is diluted to lower than 3 wt%. The color scale in Figure 16 is therefore regulated to demonstrate the portion of the injected Na-silicate that is going to be gelled (C/C₀ > 0.7).

In general, as shown in Figure 15, the required Na-silicate slug size for all the studied cases were at least 1.5 times of the one without dilution or leakoff. The profile of the injected solution changed as it propagated in the porous media, depending on the reservoir and fluid properties (Figure 16). The final profile of the formed gel was different for different studied cases; however, it was typically thick at the bottom and thin at the top, as discussed in Section 3.1.2 for gelation in sand-packs, due to the gravity segregation effect. This gel profile results in water override.

When Cases (a) and (b) are compared, it can be seen that the reservoir with permeability ratio of 100 needed considerably lower amount of Na-silicate and post-flush water (Figure 15). At a higher permeability ratio, the proportion of the injected fluid which flows through the HP layer is higher; hence the leakoff into the LP zone is lower, compared to a lower permeability ratio, which is confirmed in Figure 16a,b. However as shown in Figure 16a, most of the leaked off Na-silicate into the LP zone was diluted in the post-flush water. The small remaining gel in the LP zone in Case (a) does not considerably affect the later water flooding.

As viscosity ratio increases from 1 in Case (a) to 10 in Case (c), the required Na-silicate and post-flush slug sizes decreased (Figure 15). These values are relatively close to the Case (b), since both high permeability and viscosity ratios exacerbated the water channeling effect. Therefore, the penetrated Na-silicate into the LP layer was smaller in the Case (c) compared to the Case (a) (Figure 16c).

As shown in Figure 15, Case (d) required the maximum Na-silicate slug size among all the tested cases. Case (d) is a reservoir model which has ten times higher k_v/k_h compared to the other cases. As vertical permeability increases, the flow contribution of the water channel to the LP layer increases so larger amount of the injected fluids, including Na-silicate, leaks off into the LP layer (Figure 16d). Therefore, it is observed in Figure 16d that the post-flush water could not fully dilute the large amount of the leaked off Na-silicate in the LP zone. However, since the rear of the formed gel in the HP layer outran the front of the Na-silicate in the LP layer, the flooded water could be diverted through this gap into the LP zone and enhance sweep efficiency. The Na-silicate slug deformation due to the gravity segregation was intensified for Case (d), as shown in Figure 16d.

As shown in Figure 15, the required Na-silicate was slightly less for the reservoir without cross flow (Case (e)), compared to Case (a). The water channel expansion which occurred in the case of free cross flow (Case (a)), resulted in a higher Na-silicate penetration into the LP layer, especially close to the HP layer. This phenomenon was not the case in the reservoir without cross flow (Case (e)), as shown in Figure 16e. The gravity segregation effect, which caused gel deformation in all the previous studied Cases (a–d), did not happen in Case (e). Due to absence of flow communication between the HP and the LP layer in Case (e), the gravity of the fluid in the LP layer did not affect the gel profile.

It is important to note that, as observed in the sensitivity study above, the problem of matrix plugging is less pronounced when the injected chemical is water-soluble, such as Na-silicate, compared to the oil-soluble materials. Since the injected aqueous chemical which has been leaked off into the matrix will be dispersed into the post-flush water and diluted. However, due to the same reason, to plug a certain volume of the HP layer, it is always necessary to inject a greater volume of the water-soluble chemical, compared to the oil-soluble ones.

It was assumed that the initial water and the flooded water are sea water in the reservoir Case (a), see Table 5, while the pre-flush is LSW (with 4% relative salinity compared to sea water). Na-silicate solution was injected into the reservoir after 1 PV water flooding. The slug size of the Na-silicate solution was 0.45 PV of the HP layer. Two different pre-flush sizes, Case 1 and 2, were compared in this study. The pre-flush size was 0.09 and 0.18 PV of the HP layer for Cases 1 and 2, respectively. Figure 17 is a dimensionless graph showing concentration ratio of the injected Na-silicate slug and the relative water salinity for the two studied cases at the bottom of the HP layer as a function of dimensionless distance (x/L), where x is distance from the injector and L is the reservoir length. The results are shown at two different intervals: (I) after injecting 0.2 of the Na-silicate total volume (solid lines); and (II) when Na-silicate slug is placed at x = L/2 (dashed lines) in the HP layer. Na-silicate concentration ratio of 0.75 was taken to be the base line concentration, below which no precipitation occurs. It was also assumed that Na-silicate precipitation would not cause permeability reduction (plugging) as long as the relative sea water concentration was below 0.1 [20].

As shown in Figure 17 for Interval I, the relative front water salinity in contact with Na-silicate at C/C₀ = 0.75 was 0.06 and 0.02 for the two pre-flush Cases 1 and 2, respectively. This means that the both pre-flush cases did not cause plugging at this stage of Na-silicate injection. At Interval II, the relative front water salinity in contact with Na-silicate at C/C₀ = 0.75 increased to 0.12 for pre-flush Case 1, which shows the possibility for plugging due to precipitation in this pre-flush case. Plugging at this time may be desirable, since the gel was almost placed at the middle of the reservoir and precipitation may enhance the residual resistance factor of the treated zone. However, pre-flush Case 2 still prevented precipitation plugging in that interval.

The same study has to be done to estimate the size of LSW post-flush to avoid unwanted plugging behind the Na-silicate slug. In the case of free cross flow between the layers, it is also important to consider the water salinity in the LP layer, which is not diluted with the same rate as the HP layer. It may cause plugging in the contact area between the HP and the LP layers.

Figure 18 shows the reservoir temperature and Na-silicate slug transportation at the bottom of the HP layer as function of the normalized distance, at intervals I and II. The reservoir initial temperature was T = 120 °C, and was cooled during 1 PV water flooding of 18 °C. The major temperature decrease happened through the HP layer, where water flow was maximum. There was a temperature gradient from 18 °C, close to the injector, to about 70 °C, close to the producer, in the HP layer.

The injected slug of Na-silicate was therefore heated up as it transported in the medium. Higher temperatures accelerate the gel setting time [16], so the temperature gradient could be a controlling factor for Na-silicate gelation. When the reservoir was in the shut-in mode after Na-silicate placement, the HP layer was heated up to approximately 5 °C after four months of shut-in, as depicted in Figure 18. The temperature increase during shut-in period may help to accelerate the gelation of the placed Na-silicate.