Gelation time
Figure 2 shows the cross-linking of the polyacrylamide molecules. The criterion for the gel formation is the ability to crosslink (Tobita 1998). The gelling process involves a polymer (polyacrylamide), a reducing agent (Sodium Thiosulfate), and a metal ion (Sodium Dichromate).
The gelation mechanism for polyacrylamide in the presence of metal ions is not well understood. One suggestion is that chromium serves as a crosslinker between the polyacrylamide molecules. Another idea is that chromium forms a stable dispersion in the polymer solution which results in a high viscosity liquid or gel (Prud et al. 1983).
The reaction is hypothesized in the following sequence (Clampitt and Hessert 1974; Terry et al. 1981):
The gelation time is an important parameter that dictates at what distance from the injection well the gel will form. Therefore, the effect of reducing agent on gelation time was investigated in this study at ambient conditions (24 °C and 14.7 psia). Ten runs were performed at various concentrations of sodium thiosulfate in 2000 ppm PAM, and 1000 ppm sodium dichromate and viscosities were noted with time.
Figure 3 shows the viscosity vs. time results of one of the ten runs in which the highest yield strength was observed. The concentration of sodium thiosulfate for this optimum gelation was 2000 ppm. Other nine runs showed similar trends in way that higher gelation times were noted for a lower concentration of crosslinker.
The time zero in Fig. 3 corresponds to the time when the reducing agent (sodium thiosulfate) was added to the solution already containing the polymer and sodium dichromate. After a few minutes of lapse time, the chemical composition began to react, and viscosity began to increase significantly. However, after 25 min, at which point viscosity had reached to approximately 12.3 cp, the gelation rate jumped suddenly, rapidly, and linearly, reaching to a peak viscosity of 151.6 cp after 120 min from the start. Therefore, it is concluded that after two hours the gel was completely formed.
It is important to mention that the gelation time is usually the time required to reach an inflection in the viscosity vs. time curve, i.e., the time at which a sudden onset of sharp viscosity increase takes place. However, in this study, the gelling viscosity was initially increasing due to the cross-linking of \({\text{C}}{{\text{r}}^{{\text{+3}}}}\) ions with the polymer via the sodium dichromate group, then followed by a sharp viscosity decrease. Therefore, the gelling time for this study was defined as the time at which peak viscosity occurred.
The reason for the sharp decrease in viscosity after gelling time (reaching a peak) is not clear. It is possible that the cross-linking agent \({\text{C}}{{\text{r}}^{{\text{+3}}}}\) was responsible for the degradation of polymer gel characteristics or some other chemical resulting from the oxidation–reduction reaction. Such a behavior was reported previously by Hu 2016 (Jia et al. 2016). However, his study was for \({\text{F}}{{\text{e}}^{{\text{+3}}}}\) system and he blamed the degradation on the presence of \({\text{F}}{{\text{e}}^{{\text{+3}}}}.\) Another explanation might be that this behavior is due to excessive sodium thiosulfate, which led to a more rapid sodium dichromate reduction and a higher Cr+3 ions concentration throughout the early stages of the gelation reaction, resulting in more nucleation sites and production of weaker gels.
Yield strength
To see the effect of sodium thiosulfate concentration in a solution with 2000 ppm PAM and 1000 ppm sodium dichromate, ten concentrations of sodium thiosulfate ranging from 500 to 8000 ppm were tested for yield strength, which is an indication of the minimum shear stress required for displacing the gel. The sodium thiosulfate range of 500–8000 ppm was selected because the literature reports a minimum concentration of 500 ppm to form the gel (Batycky et al. 1982).
Yield strength was calculated using Eq. 1 by measuring the pressure drop between the inlet and outlet of the “Bead-pack flooding system” described in “Bead-pack flooding system”.
Figure 4 shows the yield strengths of the cross-linked PAM at different concentrations of sodium thiosulfate. Interesting, the trend of yield strength was quite similar to the trend of viscosity vs. time.
The yield strength started increasing at low sodium thiosulfate concentration of 500 ppm to 1000 ppm, beyond which yield strength jumped sharply and linearly peaked at 1140 psi at 2000 ppm. This increase in yield strength is attributed to a higher number of crosslinking between PAM molecules caused by a higher number of released \({\text{C}}{{\text{r}}^{{\text{+3}}}}\) ions.
At the peak sodium thiosulfate, all of the \({\text{C}}{{\text{r}}^{{\text{+3}}}}\) ions have been extracted from sodium dichromate and any further increase in sodium thiosulfate concentration caused the precipitation of the PAM. This was supported by visual observations during testing process that sodium thiosulfate concentrations beyond 2000 pm resulted in precipitation of PAM after one day. This precipitation hinders the cross-linking process thereby reducing the yield strength.
The degradation of yield strength due to precipitation starts after 2000 pm and continues till 4000 pm of sodium thiosulfate concentration, after which the reduction is negligibly small. The reason for the small degradation between 4000 and 8000 ppm is not clear.
As suggested in the published literature, a minimum concentration of polyacrylamide is required for the gel to form; and a proper ratio between the polymer and crosslinkers is needed for optimum gelling (Grattoni et al. 2001; Koohi et al. 2010). The minimum concentration of polyacrylamide needed for gelation has been suggested to be between 1500 and 2000 (Batycky et al. 1982).
We conducted our tests with 2000 ppm of polyacrylamide to be on the safe side. We also fixed the concentration of sodium dichromate and varied the concentration of sodium thiosulfate to find the optimum ratio between crosslinkers. An alternative approach could have been to fix the concentration of sodium thiosulfate and vary the concentration of sodium dichromate.
The optimum yield strength was observed at a concentration of sodium thiosulfate of 2000 ppm while the concentration of sodium dichromate was held at 1000 ppm (to keep a proper ratio between polymer and crosslinker). A similar trend should be obtained if the concentration of sodium dichromate is varied and the concentration of sodium thiosulfate is fixed at 2000 ppm, i.e., optimum yield strength would have occurred at 1000 ppm sodium dichromate concentration regardless of which one was fixed and which was varied. Using the minimal amount of sodium dichromate to form a gel is advantageous because of its toxicity and higher cost.
The yield strength data presented in Fig. 4 is not the traditional way the Oil and Gas industry uses for reservoir calculations, therefore, Fig. 5 presents a subset of this information in the form of pressure differential and permeability as noted during the brine injection after gelation vs. sodium thiosulfate concentrations. The highest differential pressure and lowest permeability were obtained at sodium thiosulfate concentration of 2000 ppm coinciding with the yield strength.
We also measured the differential pressure and calculated the permeability (using Darcy’s Law) for the brine floods before the injection of polymer gel solution and used this information only to confirm gel formation.
The reduction in permeability seen in Fig. 5 could primarily be due to gel formation as evidenced by the highest permeability reduction coinciding with the strongest gel (both at 2000 ppm of sodium thiosulfate). The second reason could be due to the precipitation of PAM as well as other components resulting from the oxidation–reduction reaction. Such precipitation, also observed in a beaker containing same polymer gel, would plug some of the pores or otherwise offer constriction reducing the permeability.
Also, the retention of PAM in the porous media by adsorption and mechanical entrapment might have contributed to this reduction as well (Agi et al. 2018; Manichand and Seright 2014; Wu et al. 2014).
Mobility reduction
The mobility is an important concept with regards to the frontal behavior of the injected fluid. As such, a plot of mobility reduction factor (MRF) as defined by Eq. 2 at ten different sodium thiosulfate concentrations is shown in Fig. 6.
The trend, of course, follows the yield strength plot but MRF gives a better feel to reservoir engineers. Just like yield strength, 2000 ppm of sodium thiosulfate concentration was the optimum for peak mobility reduction factor of ~ 6.5. With further increase in the concentration of sodium thiosulfate, MRF began to drop sharply, dropping to a value of 3 at 4000 pm. From 4000 onward, the reduction in MRF continued albeit at a much slower rate, reaching to approximately two at 8000 ppm.
No further explanation is necessary since the trends are similar to the yield strength curve and the changes taking place are governed by the same reasons as speculated with the explanation of yield strength behavior in section.
A previous study (Batycky et al. 1982) has found that mobility reduction occurs even when gels formation is not evident at low polyacrylamide concentrations and attributed this reduction to plugging of the porous media due to the precipitation of the polymer from solution. This plugging suggests that a weak or degraded gel can still be useful for EOR. In this regard, a short-lived gel offers an advantage of fewer productivity problems near the wellbore and improved mobility farther away from the wellbore.
For the research reported in this paper, all three components were premixed (as discussed in the “Sample preparation”) and injected into the bead-pack flooding system and the pressure drop was monitored to ascertain mobility reduction due to gelation.
Before starting the gelation experiments as described above, tests were performed to check if gelation will occur in the absence of Cr+3. In these tests, a mixture of polyacrylamide and sodium dichromate was injected into the glass beds under similar conditions, followed by an injection of sodium thiosulfate. No increase in pressure drop (i.e., no mobility reduction) was observed indicating that gel did not form, for the obvious reason that Cr+ 3 was not released due to the absence of reduction reaction. This suggests that gelation behavior is not the same if Cr+ 3 was not introduced to the system.