Analytical solution of polymer slug injection with viscous fingering

We present an analytical solution to estimate the minimum polymer slug size needed to ensure that viscous fingering of chase water does not cause its breakdown during secondary oil recovery. Polymer flooding is typically used to improve oil recovery from more viscous oil reservoirs. The polymer is injected as a slug followed by chase water to reduce costs; however, the water is less viscous than the oil. This can result in miscible viscous fingering of the water into the polymer, breaking down the slug and reducing recovery. The solution assumes that the average effect of fingering can be represented by the empirical Todd and Longstaff model. The analytical calculation of minimum slug size is compared against numerical solutions using the Todd and Longstaff model as well as high resolution first contact miscible simulation of the fingering. The ability to rapidly determine the minimum polymer slug size is potentially very useful during enhanced oil recovery (EOR) screening studies.


Introduction
Polymer flooding is the most widely used chemical enhanced oil recovery (EOR) technique in the world, with more incremental oil recovery attributed to this method than all other types of chemical EOR combined [21]. It is typically used in more viscous oil reservoirs (1 < μ o < 100cp) as mixing polymer into the injected water increases the viscosity of the aqueous phase and in turn reduces its mobility relative to the oil. It is thus sometimes referred to as augmented waterflooding (e.g., [2]) or water-based flooding, as the physical properties of the injected water are modified by adding the polymer. It is particularly relevant and attractive today, because oil companies are increasingly looking to develop more viscous oil fields as the fields with lighter crudes become mature.
Injecting polymer solution rather than water results in a higher shock front saturation compared to an ordinary S. A. Abdul Hamid saa1314@imperial.ac.uk A. H. Muggeridge a.muggeridge@imperial.ac.uk 1 Department of Earth Science and Engineering, Imperial College London, London, SW7 2BP, UK waterflood, reducing watercut until the polymer front breaks through, as well as improving the overall macroscopic sweep efficiency in heterogeneous reservoirs [9]. It can also help stabilize displacements for which the oil-water shock front mobility ratio is greater than 1 [5], thus reducing or preventing the degree of viscous fingering. An example of this is shown in Fig. 1. Here, the oil is 50 times more viscous than water, so the injected water front is unstable and immiscible fingering between the injected water and resident oil is observed (see Fig. 1a). Dissolving polymer in the injected water, so that the viscosity of the aqueous phase matches that of the oil, makes the polymer front stable although the leading shock front between the connate water bank and the oil may still be unstable (Fig. 1b). Figure 1c shows the average saturation profiles between the injection and production wells for the two cases.
As for other EOR techniques, the additional costs associated with polymer injection are typically higher than waterflooding. The main extra cost is the polymer itself, incurred continuously as operating expenditure (OPEX). Sheng et al. [24] reported that the average cost of polymer (excluding related costs such as processing cost) is around USD4.00 per barrel of incremental oil achieved, which can be a substantial addition to the oil lifting cost.
Costs can be reduced by injecting a fixed volume (or slug) of polymer solution, followed by water. However, care has to be taken to ensure that the polymer slug is sufficiently Fig. 1 Illustration of the stabilization of viscous fingering by polymer injection in a line drive case with an oil-water viscosity ratio of 50 and a polymer-oil viscosity ratio of 1. a Saturation distribution seen in water injection. b Saturation distribution seen in polymer injection. c Average water saturation between the injector and the producer from (a), (b). d Oil recovery curves. All shown at 0.1 PVI. All the parameters used in this simulation are as per Table 1 (a) (b) large that it maintains its integrity from the injection to the production well. Various factors including adsorption and viscous fingering can potentially destroy the slug. Adsorption means that polymer is lost progressively from the slug as it moves through the reservoir. If the slug is too small then all the polymer may be adsorbed before it reaches the production well. Viscous fingering occurs at the trailing edge of the polymer slug because the chase water is less viscous and thus more mobile than the polymer solution. Miscible viscous fingers of water tend to form and grow into the polymer slug. If these fingers reach the leading edge of the slug then the slug integrity is destroyed and the benefits of polymer injection are lost (Fig. 2). A number of authors have presented analytical solutions that can be used to estimate the impact of adsorption on a polymer slug including [3,20,22,24], and [8], but to date no-one has derived a solution to describe the impact of viscous fingering.
Very high resolution simulations are required to properly capture the growth and development of viscous fingers (e.g., [10,13]). Using the required number of grid cells is impractical, if not impossible, especially in field-scale studies. This has driven the development of empirical fingering models which capture the average behavior of a fingered front. These were originally derived for application in miscible gas injection ( [14,16,25]) but Bondor et al. [7] proposed that the Todd and Longstaff model [25] could also be used to describe the fingering of water into the rear of a polymer slug. One drawback of these fingering models, however, is that the fitting parameters in their formulation may need to be calibrated by comparison with detailed simulation.
Using such empirical models to describe fingering has enabled various authors to subsequently derive analytical solutions to describe various miscible gas injection processes. For example, Blunt and Christie [6] and Juanes and Blunt [15] have shown how it is possible to predict the behavior of water alternating gas (WAG) displacements using the Todd and Longstaff model. Such analytical models enable rapid estimation of the best ratio of water to gas to inject to maximize recovery as well as providing a means for validating numerical models. This paper presents an analytical expression to estimate the minimum polymer slug size needed to maintain the effectiveness of the polymer flood in the event of viscous fingering of the chase water case. This is obtained from a new semi-analytical solution that describes the average effects of the fingering of chase water into a polymer

(d)
Without fingering With fingering slug in 1D. We first briefly review the analytical solution of continuous polymer injection, following which the derivation describing the dynamics of a stable chase water front position as a function of time is made. These solutions are then combined and extended to capture the effect of miscible viscous fingering of the chase water into the polymer slug. This is achieved by assuming that the average effect of fingering can be represented by the empirical Todd and Longstaff model. Finally, the validity of the analytical technique is demonstrated by comparing it against its numerical equivalent as well as high resolution first contact miscibility simulation.

Analytical solution for continuous polymer injection
Let us first consider continuous polymer injection into a homogeneous 1D model so that we can understand the key features of the displacement. We assume that the system is incompressible, there is no adsorption of polymer onto the rock, and physical diffusion and dispersion are negligible. We define a normalized concentration of polymer solution in the water phase, where c p is the polymer concentration in mass of polymer per volume of water and c p,inj is the injected concentration of polymer in the same units. Using this definition μ w (C p = 0) = μ w and μ w (C p = 1) = μ p , where μ w is the water viscosity and μ p is the polymer viscosity at the injection well. As shown in Appendix A, the conservation of aqueous phase saturation S w and C p can be expressed as two hyperbolic, first order equations where F w is the fractional flow of the water phase and f p is the fractional flow of the polymer component in the water phase.
If we assume the effects of gravity and capillarity are negligible, the fractional flow of the aqueous phase F w is given by and the fractional flow of the polymer component in the water phase is given by where Q w is the flow rate of the water phase, Q p is the flow rate of the polymer component in the water phase, k rw is the relative permeability of water, μ w is the water viscosity, k ro is the relative permeability to oil and μ o is the viscosity of oil. In this paper, we assume k rw and k ro curves depend only on water phase saturation (shown in Fig. 6) and any changes in the fractional flow function (4) are due only to changes in μ w when the polymer is added to the water phase. In the absence of viscous fingering, f p = C p as the polymer solution and the chase water are first contact miscible. t d and x d are dimensionless time and dimensionless distance in the direction of flow, respectively. They are defined by the following expressions where L is the distance between the injection and production wells, A is the cross-sectional area (perpendicular to flow), φ is the porosity and Q is the injection rate. For a line drive in which polymer is injected continuously into a reservoir containing oil and immobile connate water (at saturation S wc ), the initial conditions are S w = S wc , C p = 0; for all x d If the injected water phase's viscosity remains constant (i.e., continuous water injection without polymer or continuous secondary polymer solution injection), we then have a Riemann problem in the half-plane which can be solved by using the method of characteristics (MOC), with x d /t d as the self-similar variable. Following Pope [19], the solution to this problem can be obtained graphically using two fractional flow curves, for water-oil (F w (S w , C p = 0)) and polymer-oil (F w (S w , C p = 1)) as shown in Fig. 3a. Note that we use C p = 1 here for simplicity. The following solutions are valid for any values of 0 ≤ C p ≤ 1 as long as the F w curves are correct for the selected concentration values.
The resulting solution for S w against x d is shown in Fig. 3b. Unlike the Buckley-Leverett solution for waterflooding, the profile of S w against x d is characterized by the existence of two shocks and a spreading wave. The first shock with saturation S 1 exists due to the formation of a connate water bank (C p = 0) that has been displaced by the more viscous polymer. This is followed by another shock, at which C p increases from 0 to 1, and water saturation increases from S 1 to S 2 . S 2 is found by the tangent to the polymer fractional flow curve that goes through origin  Fig. 3a, while the "jump" from water to polymer curves (S 1 to S 2 ) must satisfy the following condition which imposes the conservation of mass at the vicinity of the shock

Analytical solution for polymer slug injection
Having understood the key features of continuous polymer injection into an oil reservoir with no mobile water initially, we now examine how this is modified when we inject a slug of polymer followed by chase water. This analytical solution is complicated by the fact that there is a third discontinuity that occurs between the chase water and the trailing edge of the polymer slug, in addition to the two shocks described in the previous section. As noted in the introduction, analytical solutions to this problem have been presented by Bedrikovetsky [3], Ribeiro et al. [22], de Paula and Pires [20], Borazjani et al. [8], and Vicente et al. [26]. However, these solutions are more mathematically complex than those presented here; moreover, they focused on assessing the effects of adsorption and did not consider the effect of viscous instability.

No fingering
We first consider the case when there is no fingering of the chase water into the polymer slug. Polymer injection is stopped at time t d,slug and followed by chase water. The injection rate of the chase water is the same as the injection rate used to inject the polymer solution.
In the case of chase water injection, a further discontinuity forms at the trailing edge of the polymer slug, in addition to the shock at the front of the connate water bank and the shock at the leading edge of the polymer slug. This continuity does not travel at constant speed, unlike the first two shocks. This was first analyzed by Bedrikovetsky [3] using Green's theorem; however, we shall explain the dynamics of this discontinuity more simply here using a geometric construction in the graph of fractional flow against water saturation. This will enable us to describe the late time behavior of the discontinuity and subsequently derive an analytical expression for estimating the minimum slug size in the presence of viscous fingering in the next subsection.
The boundary conditions corresponding to the injection of the chase water are The discontinuity between the chase water and the polymer slug is found at distance x 3 from the inlet. The water saturation immediately downstream of the discontinuity is S 3 and the water saturation immediately upstream is S 4 .
Let us now find x 3 and saturation S 3 . First, select a value S 3 on the polymer-oil fractional flow curve remembering that S 2 < S 3 < 1 − S or . We then draw a line tangent to this fractional flow curve with a gradient of v 3 = dF w (S 3 ) /dS w . This line passes through the y-axis and x-axis at points A and B respectively as illustrated in Fig. 4a. Now consider Fig. 4b. The area under the curve representing the slug is given as the sum of A 1 and A 2 . For the polymer volume to be conserved (assuming no adsorption), this area must equal the injected volume of t d,slug . Hence, we have where v 2 is the characteristic velocity of the leading edge of the polymer slug.
Since for Note that when t d = t d,slug , x 3 = 0 and so F 3 = 1. This indicates that the trailing edge of the slug is at the injection well and the water saturation at the injection well is 1, as expected. As t d increases, F 3 − S 3 v 3 decreases reflecting a decreasing value of S 3 which causes F 3 to decrease and v 3 to increase (remember that S 2 the leading shock front saturation of the polymer slug remains constant). To find the limiting value of S 3 (the saturation at the trailing edge of the polymer slug) and v 3 , we let t d go to infinity in Eq. 10. This gives v 3 = F 3 /S 3 . This can only be true if S 3 has decreased to S 2 so the trailing edge of the polymer slug is traveling at the same speed as the leading front, i.e., This means that at late times, we have a rectangular saturation profile for the slug. The length of the slug, x 2 −x 3 , is simply given by At the chase water front, the value of S4 can be determined graphically from S 3 at any given time by performing a jump from polymer-oil curve to water-oil curve in Fig. 4a similar to Eq. 8 1D solution for polymer slug injection. a Fractional flow curves for water-oil and polymer-oil. The blue dashed line with a gradient of v 3 that goes through points A and B can be used to calculate x 3 using (11). The change in saturation from S 3 to S 4 can be determined by taking a jump between the two curves using a line that goes through the origin, S 3 and S 4 . b The position of slug trailing edge x 3 is determined by considering the area A 1 + A 2 which should be equal to t d,slug

Empirical model of miscible viscous fingering
The discussion thus far assumes that the interface between chase water and slug tail is stable without any occurrence of viscous instability. In reality, the interface is unstable and viscous fingering of the chase water into the polymer slug is expected as the chase water is less viscous than the polymer slug and the water and polymer solution are first contact miscible. In principle, high resolution simulation is required to model the evolution of each of the fingers; however, as noted previously, Bondor et al. [7] proposed using the Todd and Longstaff model to describe the average behavior of the water and polymer components when this fingering occurs. This is now the standard way of representing the effects of viscous fingering in commercial simulations because (a) it can be implemented relatively easily in the existing black oil simulator framework and (b) it provides a fast way of approximating the likely impact of viscous fingering of the chase water on a polymer slug. Bondor et al. [7] proposed that fingering of the chase water into the polymer slug could be treated as the fractional flow of two components of the water phase: the chase water and the polymer slug. The effective polymer slug and chase water viscosities in each grid block, μ p,eff and μ w,e , should be calculated using: where μ p is the viscosity of the polymerized water at maximum polymer concentration (C p = 1), μ w is the pure water viscosity, and μ m (C p ) is the viscosity of a mixture of pure water and polymer solution as a function of the polymer concentration. ω is a mixing parameter that can vary between 0 (no fingering) and 1 (complete mixing) but is typically set to 0.67 when there is fingering mainly based on the calibration made against experimental results made earlier by Blackwell et al. [4]. In general, this needs to be calibrated by comparison with detailed fingering simulations. The effective aqueous phase viscosity, μ w,eff , used in the 2-phase black oil model is then calculated using

Analytical solution with fingering
As noted above, existing analytical solutions describing the dynamics of injecting a polymer slug followed by chase water ignore the impact of the chase water fingering into the trailing edge of the polymer slug. We now develop an approximate analytical solution that captures this effect and show that the resulting solution can be used to estimate the minimum polymer slug size needed to ensure that it is not destroyed by fingering before it reaches the injection well. We achieve this by using the Todd and Longstaff model to represent the average effects of the fingering. To incorporate miscible fingering between the polymer slug and the chase water using the Todd and Longstaff model, we take the mass balance Eq. 3 and replace the fractional flow f p with using Eqs. 15-16. An analogous formulation for the wateroil-solvent system was used by Blunt and Christie [6] and more recently by Juanes and Blunt [14] to derive an analytical solution for WAG injection, although in WAG injection the fingering takes place at the front between the solvent (gas) and the displaced oil. They found exact solutions by simultaneously solving the conservation Eqs. 2 and 3. From Eqs. 12-13, we assume that the water phase saturation in the polymer slug is constant and given by the saturation at the leading edge of the polymer slug. We further justify this assumption with the observation that this saturation is typically high and the change in saturation across the slug, from trailing to leading edge, is relatively small. This assumption means that, we can reduce (3) to In this case, we can obtain the dimensionless position of a given mean concentration of polymer C p , from the injection well, using For example, by using ω = 0.67 in Eq. 20 allows visualization of the spreading wave at the back of the slug as shown in Fig. 5. The front of the spreading wave (C p = 1) characterizes the finger tips, which travel at a faster speed than the slug front. Its approximated velocity is given by Eventually, the finger tips will reach the slug front and cause the slug to break down. The time at this occurs when the blue line (giving the approximate characteristic speed of the leading fingers) intersects the red line (giving the characteristic speed of the leading edge of the polymer slug).  Table 1 The minimum slug size needed to ensure that the chase water fingers only just cross the polymer slug by the time the polymer breaks through is given by Further detail is given in Appendix B. Equation 22 can also be derived graphically as presented in Section 5.

Evaluation of the analytical approach
We now evaluate the above analytical solution by comparing its predictions with results from detailed simulation of the viscous fingering of chase water into a polymer slug and black oil simulation in which the Todd and Longstaff model has been implemented in the polymer options.

Methodology
To model miscible viscous fingering into the trailing edge of the slug, we used a higher order, IMPES (implicit pressure, explicit saturation), finite difference simulator developed to model the details of viscous fingering in three component, two phase flows ( [10,11]). This was originally written for miscible gas applications but was adapted to model polymer slug injection. Its ability to predict viscous fingering in miscible displacements has previously been validated by comparison with experimental results by Christie [11], Christie and Jones [12], and Al-Shuraiqi et al. [1], among others.
The commercial simulator ECLIPSE [23] and an opensource simulator called MATLAB Reservoir Simulation Tools, MRST [17] were used for the black oil simulations using the Todd and Longstaff model to capture the average effects of fingering. An identical set of equations governs both simulators, and we chose to use the fully implicit scheme in both these simulators throughout the study.
All simulations used the data summarized in Table 1, unless stated otherwise. All the symbols and nomenclature are defined in Table 2. The oil and water relative permeability curves are illustrated in Fig. 6. Grid dimensions were chosen following a grid refinement study.

The modified Todd and Longstaff model
Standard implementations of polymer flooding models in commercial simulators, such as ECLIPSE, include the Todd and Longstaff model to describe the effect of mixing between polymer solution and water and its effect on water phase viscosity. This is presumably intended to allow engineers to model the fingering of chase water into polymer as proposed by Bondor et al. [7]; however, using ω > 0 can also result in unphysical spreading of the shock front at the leading edge of the polymer slug in the absence of adsorption. The front at the leading edge of the polymer slug is no longer self-sharpening and considerably more grid blocks are then required to resolve it. This effect is illustrated for a continuous polymer flood, in Fig. 7. Figure 7a shows the polymer front obtained from 1D ECLIPSE and MRST simulations with the analytical solution for ω = 0 (no mixing between polymer and water) and ω = 1 (complete mixing). One hundred grid blocks were used in both cases. Both simulators give similar results in terms of numerical accuracy (first order) and in both cases the polymer front is smeared when ω = 1. Figure 7b shows that 10,000 grid blocks are needed in this case to obtain a shock front close to the analytical solution. Clearly, this is impractical for field-scale simulations. One solution is to use higher-order solvers ( [18,19]), but these are more computationally expensive. Alsofi and Blunt [2] used a simpler method in streamline-based simulator, whereby the weighted average fractional flow is used to segregate the flow between the regions with and without polymer.
In this study, we adapted MRST using an approach comparable to the method proposed by Alsofi and Blunt [2]. At every time step, we calculate the maximum value of polymer concentration, C p,max reached in each of the grid blocks since the beginning of injection. We then modify the Todd and Longstaff mixing parameter using ω = ωC p,max (23) where ω is the input value of the Todd and Longstaff mixing parameter. This means that the mixing parameter is zero in cells that have never seen polymer and reduced in cells that see polymer increasing (as the polymer slug advances) but is equal to the input value at the trailing edge of the polymer slug. This strategy was found to give better resolution of the leading polymer front in MRST as shown in Fig. 8.

Results-no viscous fingering
We first verify that the black oil simulators can reproduce the analytical solution derived in Section 3.1 for polymer slug injection for the case without viscous fingering. Figure 9 shows the comparison between the analytical solution and the predictions of the modified MRST using ω = 1 for a slug volume of 0.4 PV. Other input data is as per Table 1. A very high resolution ECLIPSE model (DX = 10000) is also shown here.
The simulators correctly capture the location of the trailing edge x 3 and the relatively small jump in saturation from S 3 to S 4 at x 3 . The immobile oil region, formed by the locus of chase water front saturation S 4 , is also observed.

Results-viscous fingering
We now evaluate the ability of the new analytical model to predict viscous fingering of the chase water into the trailing edge of the polymer slug by comparing its predictions with those obtained when we model the fingering explicitly. Here, we consider two cases, μ p /μ w , of 10 and 50. We keep μ p = μ o in all cases so that the potential impact of immiscible fingering at the slug front can be ruled out. The polymer slug size is 0.5 PV. We also compare these two predictions with those obtained from the modified MRST using the Todd and Longstaff model with ω = 0.67. Figure 10 shows that there is very good agreement between the three approaches suggesting that the assumptions made in the derivation of the analytical model are appropriate. In particular, the time when the finger tips reach the polymer front is successfully predicted analytically.

Minimum slug size computation
We now show how the analytical model can be used to calculate the minimum polymer slug size graphically that will maintain its integrity between injection and production wells even if there is fingering of the chase water into the trailing edge of the slug. The smallest slug size to minimize the breakdown of the slug by fingering should be selected such that the finger tips reach the production well (x d = 1) at the same time as the polymer front. This can be graphically determined by drawing a line with a gradient of v tips given in Eq. 21 that meets x 2 at x d = 1. The green line in Fig. 11 illustrates this. Then, t d,slug,min can be directly determined from the value of the x-intercept. Assuming ω = 0.67, we found that t d,slug,min = 0.52 PV.
To verify this, we ran the FCM simulation for t d,slug of 0.3, 0.52, and 0.65 PV. The results are shown in Fig. 12. As expected, t d,slug = 0.3 PV is too small and viscous fingering of the chase water causes the slug to break down in the middle of the reservoir. This is reflected in the oil recovery plot in Fig. 12d which indicates that the polymer breakthrough at the production well occurs rather early, at around t d = 0.5 PV. We also observe that the recovery is less efficient as the increase in oil is very gradual post polymer breakthrough. The recoveries obtained for t d,slug = 0.65 PV and t d,slug = 0.52 PV are found to be almost identical, which means that, in this case, the minimum slug size needed to ensure fingering of the chase water does not destroy the slug is t d,slug = 0.52 PV. Fig. 13 Optimum slug size as a function ω and viscosity ratios. All data used in this calculation is from Table 1 The minimum slug size needed to ensure fingering of the chase water does not completely break down the polymer slug for various values of ω and viscosity ratios calculated using Eq. 22 are shown in Fig. 13. We can see here the expected trend showing that high oil viscosities or low ω values necessitate the use of larger slug size. Generally in polymer flooding, polymer viscosity needs to be high enough that the oil can be swept more efficiently than waterflooding. However, interestingly, too high a polymeroil viscosity ratio, μ p /μ o , has an adverse effect instead-a large slug size is required as the chase water tends to finger more through the polymer slug as we can see in Fig. 13b.
Although we need to consider many parameters to calculate the required slug size, for moderate values of ω of say around 0.4 to 0.8, Fig. 13 suggests that t d,slug,min of around 0.5 to 0.6 PV is a good first approximation that may be used in the field.

Conclusion
We have investigated the fingering of chase water into a polymer slug during secondary polymer flooding in the absence of adsorption using a mixture of numerical simulations and analytical approaches. Both detailed numerical simulations, describing the fingers explicitly, and black oil simulations using a Todd and Longstaff model to represent the average effects of the fingering were performed. We have reviewed the existing analytical solutions that predict both continuous polymer flooding and a slug of polymer followed by chase water in the absence of fingering.
Existing solutions describing the injection of chase water following a polymer slug assume a stable interface between chase water and polymer slug. A graphical solution based on Welge analysis was presented in Section 3.1 and compared successfully against numerical simulation. We showed analytically that at late times the water saturation throughout the polymer slug tends to the saturation of the leading shock of the polymer slug.
We have extended this analysis to obtain an approximate method for predicting the growth of the fingering of chase water into the polymer slug due to fingering, in the absence of adsorption. This analysis provides a simple analytical expression that can be used to estimate the minimum polymer slug size needed to ensure that it is not destroyed by fingering of the chase water before polymer break through.
Comparison of detailed numerical simulations of the fingering of chase water with black oil simulations using the Todd and Longstaff model to represent the average effects of the fingering have shown that a value of ω = 0.67 can be used, for oil-water viscosity ratios of 10 and 50. However, due to the empirical nature of the Todd  Modified ω and Longstaff model, this value may not work for cases with heterogeneous reservoirs, very high viscosity ratios, or polymer-oil viscosity ratios that are less than 1. We expect calibration of ω is required for more general problems.
When ω is provided, the optimum slug size can be rapidly determined using the approximate analytical solution. Such results can potentially be used during EOR screening or feasibility studies, during which only order of magnitude estimations are required and more accurate but highly expensive computational study may not be necessary. We expect that the analytical solution presented can be extended to include the effects of adsorption and possibly to estimate flow behavior when polymer is injected subsequent to a waterflood.
With no fingering, the back of the slug will have approximately the same velocity as the slug front (as discussed in Section 3.1) so v 3 = v 2 . Hence, we have To take the fingering into account, we multiply x 3 in (B.2) by df p /dC p . The fastest wave occurs when C p = 1, shown as the upper blue line in Fig. 5. At the production well, we have The minimum slug size needed to ensure that the chase water fingers only just cross the polymer slug by the time the polymer breaks through is found by substituting (B.1) into (B.3)