3.1. Sensitivity to Permeability Range
Before presenting the detailed simulations of the 2D experiments, a number of sensitivity calculations are performed to demonstrate the rationale for the permeability range used in this study. These examples all use the input parameters derived for the 2000 mPa.s case, which is presented in detail later in this paper.
The permeability field is numerically re-scaled to a range of k
min–k
max values using Equation (6) such that the permeability range moves from the base case of 1:1000 to 1:2. Rescaling the permeability field in such a way maintains the exact permeability field
structure through all sensitivity calculations. The k
min and k
max values used have been selected to maintain an average permeability of ~3 D.
The resulting finger patterns prior to water breakthrough (~0.04 PV) are shown in
Figure 2 along with the respective permeability field to reinforce that the changes in finger patterns are solely due to the permeability values themselves. Similarly,
Figure 3 shows oil recovery, water cut, and pressure drop for each of the cases.
It is clear from
Figure 2 that the permeability range utilised in the simulation grid can have a significant impact once the ratio of k
min:k
max is less than 1:5. A lower ratios the finger patterns begin to become more linear and by a ratio 1:2 show very little of the dendritic behaviour observed during highly unfavourable displacement. However, on examination of
Figure 3 it is clear that while k
min:k
max affects the finger patterns, little impact is observed on the oil recovery, water production and pressure drop. Of course, as the fingers blunt the front is more stabilised and the water breakthrough is slightly delayed at low k
min:k
max ratios. This in fact simplifies the simulation process when simulating the unstable immiscible displacement processes detailed here—a precise understanding of the
exact permeability distribution is not required, but rather through observation of the finger patterns the permeability range can be deduced., i.e., the correct permeability distribution is the one that gives representative finger patterns. Indeed, it is possible to match the production data and pressure drops with a wide range of permeability values if in situ imaging is not available as a constraint.
As a result of this sensitivity calculation and those previously performed in the literature, a 500 × 500 grid, λD = 0.03 and kmin–kmax = 1:1000 have been used throughout as it provides the most representative finger patterns when compared to the experiments.
3.2. Displacement of the µo = 2000 mPa.s Oil
The simulation match to the experimental data for the 2000 mPa.s case is given in
Figure 4.
It can be seen from
Figure 4 that the approach used here can be utilised to obtain an excellent match to the experimental oil recovery, water cut, and differential pressure observed in the experiment. It should be noted that there is some mismatch in differential pressure at the end of the water flood (~1.5–2.25 PV) however this remains in the limits of experimental error. Importantly, both the timing and magnitude of polymer response are very well captured here. The large, quick response to the injection of a moderate viscosity polymer solution (µ
p = 60 mPa.s) is in line with the viscous crossflow mechanism proposed by Sorbie and Skauge (2019) [
24], as the displacement is still operating a strongly unfavourable viscosity ratio, (µ
o/µ
p) = 2000/60 ~ 333, although there is a bank of connate and injected water in front of the polymer bank.
The simulated finger patterns are compared with experimental observations in
Figure 5. During the polymer flood, the experimental images are produced by subtraction of a reference image of the end of the water flood (image at 2.26 PV in
Figure 5)—blue indicates an increase in water saturation whereas red indicates an increase in oil saturation.
There is good agreement between the simulated finger patterns during the early stages of the water flood in that approximately the correct number of fingers are observed and the dominant fingers show both spreading and tip splitting. At later stages (>0.14 PV) the simulated fingers are slightly more disperse than the experimental images. It may be a limitation of the simulator/methodology not able to catch the collapse of fingers into channels of high-water saturation. These channels formed are important for initiation of the viscous crossflow mechanism and the resultant acceleration of oil production by polymers. However, although they are not perfectly resolved in the late time finger images, they do result in the correct amounts of incremental oil (IR) recovered by the polymer in this case (see
Figure 4) and in all later cases.
As the simulation moves to displacement by the polymer viscosified water phase, the simulation provides a good approximation of the saturations. At 0.11 PV there is a somewhat reduced oil bank vs. the experimental image, however by 0.25 PV the simulation is in line with the experiment distributions. The simulation captures the building of an oil bank along with the unstable front of the aqueous polymer phase (which is clearly still not completely stable).
The input parameters required to achieve this match for the 2000 mPa.s oil case are detailed in
Table 3 and the resultant relative permeabilities, fractional flow and total mobility functions are shown in
Figure 6.
From the fractional flow in
Figure 6 the water saturation at the shock front of the finger (
) can be calculated from the tangent as,
Swf = 0.19 which has a corresponding value of
= 1.5.
3.3. Displacement of the µo = 7000 mPa.s Oil
Figure 7 shows the simulation match for the 7000 mPa.s oil viscosity case.
As with the 2000 mPa.s case, very good agreement is observed between simulation and experiment for recovery factor, water cut, and pressure drop. The extremely strong response to a modest increase in displacement fluid viscosity (1 to 60 mPa.s) relative to the defending fluid viscosity (7000 mPa.s) is well captured and supports the proposed viscous crossflow mechanism.
Figure 8 compares the simulated finger patterns against the X-ray saturations observed in the experiment. As with the 2000 mPa.s case, the polymer flood images have been generated via subtraction of the end of water food image.
As with the 2000 mPa.s case, the early portion of the water flood for the 7000 mPa.s oil shows representative finger patterns when compared to the experimental images. As the flood develops, a more disperse finger pattern is obtained in the simulation vs. the experiment, however as noted above, this may be in part due to the image threshold/sensitivity to the very lowest water saturations.
Figure 9 demonstrates the significant change in observed finger pattern with a change in threshold of just 0.07 saturation units.
During the injection of polymer viscosified water,
Figure 8, the distribution of fingers, oil bank and instability of the polymer bank is well captured at all stages of the polymer flood. Some delay in the formation of the oil bank vs. the experimental images is observed despite the good agreement observed between the simulation and the effluent samples,
Figure 7.
LET parameters used to perform this simulation are given in
Table 4 and the resultant relative permeabilities, fractional flow and total mobility functions are shown in
Figure 10.
In the case of 7000 mPa.s oil, the fractional flow gives = 0.094 and a corresponding value of = 1.14.
3.4. Displacement of µo = 412 and 616 mPa.s Oils
Upon examination of the experimental finger patterns observed in the cases of 412 and 616 mPa.s oils, it was clear that some larger scale heterogeneity features must be present, since the flows clearly appeared to “avoid” certain regions of the slab, and “prefer” others. As a result, it was only possible to obtain similar finger patterns in the simulations when some degree of structured heterogeneity was included in the model superimposed upon the random correlated permeability field. As shown above, this was not necessary for the 2000 and 7000 mPa.s simulations, which only required a single random correlated permeability field.
The observed flows for the water injection stages of the 2D slab floods directly suggested the simple changes to the permeability field required. The resulting modified permeability fields are shown in
Figure 11. The same base permeability field is used, as in
Figure 1, however in the 412 mPa.s case the center 50% of the grid was reduced in permeability by half. While the permeability in the 616 mPa.s case is reduced by half on the left 50% of the grid. Recall that the 412 mPa.s experiment was performed in a smaller slab (15 cm × 15 cm) than the other 3 experiments (30 cm × 30 cm), and therefore a coarser grid was used (250 × 250 cells) to maintain the same grid dimensions across the different simulations. We originally regarded making these changes as being somewhat arbitrary and rather unsatisfactory, and considered omitting them from this paper. However, the water and tertiary polymer displacement simulation results are reported here for two reasons, (i) they match the experiments extremely well in all respects, as shown below, and more importantly, (ii) they actually demonstrate the presence of a “double” viscous crossflow (VX) mechanism which leads to a large response to polymer for each of these 2 cases (explained below).
The simulation matches to the experiment using the grid modifications described above are shown in
Figure 12. The oil recovery number in
Figure 12 (summarised in
Table 2) indicate incremental recoveries due to the tertiary floods (IR) of ~100% in each case. As we noted above, this is an extremely high value of IR for any EOR process. However, as for the other higher oil viscosity cases, the simulations using the modified permeability model gave exactly the same quantitative recovery responses. However, as we explain below, there is additional mechanism at work in these cases due to the assumed (and observed) permeability heterogeneity.
It can be seen that a very good (but not perfect) match between experiment and simulation is obtained for both floods for the oil recovery, water cut, and pressure profiles. The corresponding finger patterns observed in the simulation vs. experiment are presented in
Figure 13 for the 412 mPa.s case and
Figure 14 for the 612 mPa.s case. By studying the animations of the evolution of the finger patterns, the water/oil saturations, and the oil banking, it became clear what mechanisms led to the very high IR values for the polymer response described above (IR~100%). The animations of water saturation against time are supplied as
Supplementary Materials with this paper and may be downloaded from the journal website (
Videos S1–S4).
Analysis of the 412 mPa.
s case: The finger patterns during water flooding of the 412 mPa.s oil is presented in
Figure 13 show excellent agreement with the experiment with some acceleration in propagation of the central fingers vs. the edge fingers at the end of the water flood. As in the previous simulation cases, the finger pattern at the end of the water flood is slightly more dispersed for the dominant fingers than in the simulation. However, even well after water breakthrough (at 1.03 PV), very little injected water has entered the low permeability central channel. This is because the higher permeability channels on either side are full of low viscosity (1 mPa.s) water and the bypassed lower permeability central channel contains much more viscous (412 mPa.s) oil. It is this situation which maximises bypassing and reduces any possibility of additional water injection displacing the oil in this central region by continued water injection. However, this situation does increase the target oil for the subsequent polymer injection, as discussed below.
As the viscosified polymer flood begins, the results in
Figure 13 show two important effects in
both the experiments and the simulations, as follows: (i) the finger patterns change and viscous fingering is now observed in the central low permeability channel, and (ii) very clear oil banking occurs and some of this is due to larger scale crossflow of oil from the central low permeability region into the swept regions on either side of the system. As with the experiment, at the end of the flood the entire slab is swept to an almost constant water saturation. The reason for the appearance of fingering in the central region is that the polymer banks the already injected water and the connate water, and it is this which causes the fingers in this central region of the system. The simulations reproduce all of this behaviour extremely well, and it is our observations from the corresponding animations that reveal that
two viscous crossflow (VX) mechanism are at work here at different scales in this system, viz. (1) firstly the polymer mechanism at the scale of the individual fingers is causing banking of oil by a local VX mechanism (as it did in the 2000 and 7000 mPa.s cases above); and (ii) there is an
additional “layer to layer” VX mechanism causing displacement of the oil in the low permeability central zone. This latter “layer to layer” VX mechanism is the one that has been well described in the literature for many years (Sorbie, 1991 [
23] and many references therein). Superimposed on the VX mechanism is, of course, the direct displacement “extended Buckley-Leverett” mechanism as described above in the introduction, but this is the weaker mechanism in terms of its contribution to IR. In summary, it is this “double VX” mechanism due to the viscous polymer that is additionally contributing to this very large IR response.
It is fortuitous that this occurs in this 2D slab flood, since it has brought to light an experimental case showing the “double VX” mechanisms which has helped us to explain the response to polymer. However, it also implies that we must take some care in comparing the results for this case (and the 616 mPa.s case below) directly with the 2000 mPa.s and 7000 mPa.s results, because of the additional factor in these cases, i.e., the heterogeneity in the 2D slab systems.
Analysis of the 616 mPa.
s case: During water flooding, the simulated finger patterns for the 616 mPa.s case presented in
Figure 14 show good agreement with those obtained experimentally by in situ imaging of the 2D slab floods. In this case, the permeability field has a higher permeability channel on the right half of the system, as described above. In this high permeability we observe one larger, splitting finger in the simulations compared with the two more stable fingers observed in the experiment. This is a relatively minor deviation that is simply due to the specific random permeability field used in this set of experiments. However, both the simulation and experiment show clear preferential bypassing on the right of the model leaving most of the bypassed oil on the left of the model. As with the 3 previous displacement cases, by the end of the water flood the primary fingers have become more dispersed than those presented in the experimental images, however, as mentioned above this could be due to the processing of the experimental images.
During polymer viscosified water injection, crossflow from the low permeability zone (left) to the high permeability zone (right) can be seen at 0.06 PV and by 0.13 PV (see
Figure 14) which has resulted in an even oil saturation across all flooded areas. Following this, the slab is systematically swept in line with the experimental measurements. Again, like the 412 mPa.s oil displacement, the response of the 616 mPa.s oil recovery to polymer can be analysed in terms of the “double” viscous crossflow (VX) mechanism as descried above.
The LET parameters for both floods (µ
o = 412 mPa.s and 616 mPa.s) are shown in
Table 5 and the resulting relative permeabilities, fractional flows and total mobilities are shown in
Figure 15.
From the fractional flow tangent, it can be calculated that the is 0.40 and 0.35 for the 412 mPa.s and 616 mPa.s cases, respectively, while the is 6.36 and 4.97, respectively.