Experimental study of the growth of mixing zone in miscible viscous fingering

Displacement of a more viscous fluid by a less viscous fluid leads to flow instabilities resulting in the formation of fingers of the less viscous fluid. The onset and evolution of instabilities is referred to as viscous fingering.^1–4 Viscous fingering has been a subject of extensive study over the past years because of its applications in many processes which include flow through porous media,⁴ secondary and tertiary oil recovery, ^5–7 proppant placement in hydraulic fractures, ^8–10 and flowback from hydraulic fractures.¹¹ In case where the displaced and displacing fluid are miscible with each other, the formation of instabilities is referred to as miscible viscous fingering. The process of miscible viscous fingering can be modeled by solving the following equations:

where u is the total flux of the fluid per unit area, c is the volume fraction of the solvent, and μ(c) is the viscosity of the homogeneous fluid mixture with a solvent volume fraction c. D is the diffusion coefficient and p is the pressure in the fluid. In this model, diffusion is assumed to be isotropic which might not be the case in a real situation especially in porous media. If the effect of gravity is neglected, dimensional analysis of the problem suggests that two dimensionless parameters, Peclet number and viscosity ratio, determine the behavior of the model Peclet number, Pe = uL/D is defined as the ratio of time scale of diffusion to the time scale of convection. Here, L is a characteristic length scale, taken to be equal to the length of the flow cell in the current work. Viscosity ratio is the ratio of viscosity of displaced fluid (μ₁) to the viscosity of displacing fluid (μ₂). Equations (1)–(3) neglect the effect of gravity, but three-dimensional simulations by Vanaparthy and Meiburg¹² have shown that gravitational forces play a role in determining the shape of displacement in simple horizontal flow geometries. Gravitational forces can modify the flow around the tips of viscous fingers affecting effects such as tip-splitting.¹² 

A vast body of theoretical, numerical, and experimental work devoted to understanding various aspects of the instabilities is available in the literature.^13–17 It has been shown by numerical simulations that viscous fingers form at high Peclet numbers and width of fingers is small. Accurate numerical solution is costly at high Peclet numbers and it is difficult to reproduce the detailed fingering pattern. Numerical simulations by Zimmerman and Homsy¹³ and Yang et al.¹⁸ have shown that it is simpler to describe the concentration of solute averaged across the fingers. This is particularly useful in linear geometries where pressure gradient is applied in one direction. In such cases, the mixing zone is an important quantity to determine the extent of mixing. A schematic illustrating the mixing zone is in Figure 1. The mixing zone is defined as the length where the local concentration of the injected fluid, c, varies from 0 (corresponding to the initial fluid) to 1 (corresponding to the injected fluid).

The spreading and growth of the mixing zone is an important issue that still remains unresolved. Several empirical models are available for the evaluation of mixing zones in unstable, miscible displacements.^19–21 Koval’s prediction¹⁹ of the growth of the mixing zone in a homogeneous medium is based on the following equation:

where $c ¯$ is the volume concentration of solvent averaged across the fingers, U is the total flux of fluid, and M_e is the effective viscosity ratio defined by

where M is the ratio of viscosity of displaced fluid (μ₁) to the viscosity of displacing fluid (μ₂). M_e is the same as the Koval factor, K¹⁹ for a homogeneous medium. The above equation was derived for hydrocarbon and solvent mixtures which follow the fourth-root (or quarter-power) mixing rule

μ_HC is the viscosity of the hydrocarbon, μ_S is the viscosity of the solvent, μ_mix is the viscosity of the mixture, and ϕ is the volume fraction of the solvent. The value of 0.22 was obtained by fitting the model with the recovery data of Blackwell et al.²² for miscible displacements in sand packs with viscosity ratios below 150. The Koval model does not have a rigorous derivation and should be considered as an ad-hoc fit to experimental data. Numerical simulations of Tan and Homsy,²³ Yang et al.¹⁸ and Booth²⁴ and fluid displacement experiments in rock cores by Tchelepi et al.²⁵ are in agreement with the Koval model; that when the Peclet number is large, the mixing zone grows linearly.

The mixing zone velocity can be defined as the rate of change of mixing zone length and is expressed by the following equation for Koval’s model:

The equation predicts that the mixing zone velocity is directly proportional to (M_e − 1/M_e). The growth rate of the mixing zone agrees with the experimental observations of Wooding.²⁶ Yortsos and Salin²⁷ used the approach by Menon and Otto²⁸ and concluded that the mixing zone velocity is bounded by (M − 1)²/(MlnM). The empirical models for Todd and Longstaff²⁰ and Fayers et al.²¹ give a reasonable description of the internal structure of viscous fingering patterns with possible inclusion of gravity segregation. These models not only match the growth rate of mixing zone given by Koval’s model but also predict the pressure drop through the fingering region and allow for fully three-dimensional solutions. The model by Todd and Longstaff,²⁰ similar to Koval’s model, has a single matching parameter, whereas Fayers et al.²¹ introduced a “fingering function” for the matching. For reservoir engineering applications, the model by Todd and Longstaff²⁰ is widely used.

The objective of this work is to quantify the growth of mixing zone over a wide range of viscosity ratios with an attempt to extend the range of Koval’s model to viscosity ratios of 1225. Experiments are performed in a rectilinear Hele-Shaw cell for the injection of water into glycerol solutions at high rates. The mixing zone velocity and fingertip velocity are quantified for viscosity ratios up to 1225. Results are presented for the first time to show that the mixing zone velocity and fingertip velocity initially increases with viscosity ratio up to viscosity ratios of 340 and reaches asymptotic values at high viscosity ratios. The Koval model does not capture the plateau in the velocities.

Figure 2 shows a sketch of the Hele-Shaw cell used in the experiments. The cell is made of Plexiglass, and the flow channel is 84 cm long and 5 cm wide. The walls are smooth and parallel to each other within the precision of the spacers, which is 1 ± 0.02 mm. Figure 3 shows a sketch of the experimental setup. The setup consists of two 500 ml Isco syringe pumps. Pump 1 is used to fill the Hele-Shaw cell with the displaced fluid (glycerol solution), pushing the liquid against gravity with the cell kept in an inclined position. This is done to avoid any air bubbles in the cell. An accumulator with a piston is used to pump the displaced fluid to the cell. This is done to prevent the pump from coming in contact with viscous fluids. Once the cell is filled with the displaced fluid, valve (a) is switched to the other position, and the displacing fluid (water) is injected into the cell with Pump 2. A three-pronged manifold splits the displacing fluid (water) stream into three parallel streams that enters the cell from the three sides. A water-soluble dye is added to the displacing water to increase the contrast between the two liquids. The cell is kept in a horizontal plane in all the experiments to avoid any gravitational effects in the direction of flow. Two long light sources are kept alongside the length of the Hele-Shaw cell to obtain uniform illumination. Upon the injection of the displacing fluid, a high definition video camera is used to record the experiment. The camera is placed over the top of the cell using a pair of tripods and is moved along the length of the cell during the experiment. A measuring scale is kept alongside the Hele-Shaw cell while recording the video. Similar experimental setup was used to perform the viscous fingering experiments in viscoelastic fluids.²⁹ 

An image analysis tool Tracker 4.0³⁰ is used to analyze the movement of fingers and to calculate the mixing zone length as a function of time. The x-position of the finger front (where the cross-section averaged injected concentration of injected water increases to just above zero) is tracked as a function of time. The x-position of the point where the cross-section averaged injected concentration of injected water increases to one is also tracked at the same time steps. The difference between the two x-positions gives the length of the mixing zone. The measurements are discussed in detail in Sec. IV. At least two measurements are made for each reported mixing zone length under a unique set of conditions to ensure reproducibility.

Glycerol solutions with viscosities ranging from 1 to 1225 cP are used in the displacement experiments. Steady shear-viscosity measurements are made using the ARES rheometer using a double wall concentric cylinder fixture (inside cup diameter: 27.95 mm, inside bob diameter: 29.50 mm, outside bob diameter: 32.00 mm, outside cup diameter: 34.00 mm, bob length: 32.00 mm). Ten points per decade are measured. All the solutions have a constant viscosity (within 1% of the zero shear viscosity) over the range of shear rates of 0.1–800 s⁻¹. Figure 4 shows the viscosity of the glycerol solutions, used in the experiments, as a function of the glycerol weight percentage. The viscosity readings using the rheometer are repeatable with a negligible variance. The viscosity prediction using the fourth-root mixing rule (Eq. (6)) is also shown. It can be observed that the water-glycerol mixtures follow the fourth-root mixing rule.

In order to ensure the homogeneity of the Hele-Shaw cell, a unit displacement (water displacing water) is performed. The displacing water is colored with the dye and is injected at a rate of 12.35 ml/min (highest injection rate in any of the subsequent experiments). Figure 5 shows snapshots of the displacing water front at different times. A piston-like displacement at the interface of the two fluids is observed. This confirms that the cell walls do not cause any instability, and any finger formation is due to the viscosity contrast between the two fluids (in absence of gravity). The constant velocity of the interface also helps in confirming that the spacing between the walls is constant (equal to 1 mm) through the length of the cell.

Displacement of the glycerol solutions is performed at three different flow rates: 12.35, 4.69, and 1.28 ml/min. The mutual diffusion coefficients between distilled water and water-glycerol solutions are calculated using the relation proposed by D’Errico et al.³¹ The flow Peclet numbers (Pe) are then calculated using the following equation:

In the above equation, D is the mutual diffusion coefficient and x₂ is the mole fraction of glycerol in the water-glycerol mixture. U is the velocity of flow displacements, D is the diffusion coefficient, and b is a characteristic length scale. For calculations, the characteristic length is taken as the gap width of 1 mm. The velocity, U, is calculated by dividing the injection rate by the cross section area of channel. Table I lists the diffusion coefficients, and the Peclet numbers at the three flow rates, for all glycerol solutions. The Peclet numbers are high (>10³) for all the displacement experiments performed in this study. Based on the high Peclet numbers, it can be assumed that all the displacements are convection dominated and longitudinal diffusion can be neglected.

Figures 6–11 show the sequential evolution of the interface between water and glycerol solution at an injection rate of 4.69 ml/min at viscosity ratios of 50, 79.5, 177.5, 463, 665, and 1225, respectively. As seen in these figures, initially a large number of fingers originate from the interface between the two fluids. With the progress of the displacement, the fingers become longer and wider, and their number reduces. Differences are observed in the viscous fingering patterns in different viscosity glycerol solutions at the same flow rate. At lower viscosity ratios, multiple fingers are observed to grow parallel to each other, signifying reduced shielding at low viscosity ratios (up to 79.5). Shielding is the process by which growth of a finger retards the growth of adjacent trailing fingers.^4,32 At higher viscosity ratios, shielding is more pronounced. An illustration of increased shielding can be observed in Figure 9. At the snapshot at 15 s, there are two fingers close to each wall of the Hele-Shaw cell. At later times, i.e., 53 s, it is observed that the finger closer to the wall with the scale retards the growth of the finger close to the other wall.

Spreading is the mechanism by which a finger becomes wider as the displacement continues. Coalescence is the mechanism by which the tip of one finger merges into the body of another adjacent finger. Spreading and coalescence are observed in displacement in all the glycerol solutions. The degree of coalescence increases as viscosity ratio increases, i.e., the fingers turn at wider angles to merge into the body of adjacent fingers. An illustration of coalescence can be observed in Figure 10 in snapshots at 21 s and 39 s. At the snapshot at 21 s, two fingers in the middle of the channel are moving parallel to each other. At a later time, 39 s, it is observed that the trailing finger mergers into the body of adjacent finger by turning an angle of approximately 60°. Other instances of coalescence can be observed in Figure 11 where the finger in the middle of channel splits into two (21 s), and after splitting, each branch coalesces into the corresponding side finger (42 s).

Tip splitting is the mechanism by which the tip of a finger splits into two or more fingers.^23,26 It is observed that at low viscosity ratios of 50 and 79.5, the fingers do not exhibit prominent tip splitting features. As seen in Figure 6, the only tip splitting observed is shown in snapshots at 63 s and 68 s. The tip splits into two and both fingers continue to grow. This kind of tip splitting is referred to as even tip splitting. At the viscosity ratio of 79.5 (Figure 7), none of the fingers split and the only phenomenon observed is merging/coalescence.³² However, at higher viscosity ratios, all the dominant fingers exhibit tip splitting. At a viscosity ratio of 177.5 (Figure 8), the fingers split into two, and after splitting, one finger shields/retards the growth of other. This mechanism has been referred to as uneven tip splitting.³³ The growing finger further splits into two, and the process repeats. This observation is qualitatively consistent with the numerical simulations of Islam and Azeiz.³³ As the viscosity ratio is further increased, the tip splitting patterns become complicated with finger tips splitting into three or more fingers. Instances of finger tips splitting into three and four fingers can be observed in Figures 10 and 11. In Figure 10, at 18 s, the finger in the middle of the channel splits into three fingers, one of which grows further and shields the growth of the other two. The fingertip close to the wall at 21 s splits into four fingers. One of these fingers grows and further splits into three. Similar observations are made at viscosity ratio of 1225 (Figure 11). To summarize, a transition from no tip splitting to prominent tip splitting is observed as the viscosity ratio between the fluids is increased.

Using Tracker 4.0, the length of the mixing zone (as described in the schematic in Figure 1) is calculated at different times of evolution of the interface for all the fluid displacements. Figure 12 shows the mixing zone length as a function of time in 97% glycerol solution (viscosity ratio of 665) at two flow rates. A linear increase in the mixing zone length with time is observed in all the experiments. This observation is in agreement with numerical studies^23,18,24,27 and experimental studies^25,26 from the literature. The mixing zone velocity is calculated from the slope of the straight line.

Figure 13 shows the mixing zone velocities versus the viscosity ratio in all the glycerol solutions at injection rates of 12.35, 4.69, and 1.28 ml/min. The error bars on the data points represent the variation between two and three independent experimental runs under the unique set of conditions. The absence of error bar on any data point suggests that the error is smaller than the marker size. In the three cases, an increase in the mixing zone velocity with viscosity ratio up to a value of 343 is observed. Above this viscosity ratio, the mixing zone velocity plateaus. This suggests that the rate of growth of the mixing zone does not change above this viscosity ratio. This observation is not captured by the Koval’s model which, as mentioned before, is an empirical fit to experimental data for viscosity ratios below 150.

Figure 14 shows the mixing zone velocity normalized with the injection velocities (injection rate/cross section area) at the three injection rates. It is observed that for the three injection rates the normalized mixing zone velocities converge into a single curve. This curve is a relation between the rate of growth of the mixing zone and the viscosity ratio of two fluids, in a dimensionless form. It is also worth mentioning that the Peclet number at a given viscosity ratio is different at the three different injection rates. The plot in Figure 14 shows that the dimensionless growth rate is independent of the Peclet number. This is consistent with the previous experiments and simulations by Tan and Homsy³⁴ that normalized mixing zone velocity (u/U) = 1.5 for a viscosity ratio (M) = 20. Experiments by Tchelepi et al.²⁵ in millstone cores show a normalized mixing zone velocity, u/U = 0.6 at viscosity ratio (M) = 3.1 which is consistent with the current experimental results.

Equation (7) can be rearranged in the following form:

The right hand side of this equation is the normalized mixing zone velocity shown in Figure 14. Using the experimental data of the normalized mixing zone velocities, the Koval factor, M_e, is calculated from the above equation. It is compared with the Koval factor, M_e, calculated from Eq. (5). The comparison is shown in Figure 15. It is observed that M_e values from the experiments are lower than those predicted from Eq. (5). It can also be observed that M_e calculated from the experimental data plateaus at viscosity ratios above 343 which is not captured by the Koval’s formulation.

As mentioned in the Introduction, the value of 0.22 in Eq. (5) was derived from fitting the experimental data of Blackwell et al.²² To match our experimental data for viscosity ratios below 343, the factor ϕ, i.e., the volume fraction of solvent (Eq. (6)) is adjusted. This is done on the basis that the water-glycerol mixtures obey the fourth root mixing rule. A ϕ factor of 0.094 instead of 0.22 is observed to fit the mixing zone velocity data below viscosity ratios of 343, i.e., the following equation combined with Eq. (7) are observed to fit the data:

The comparison of the data and the above equation is shown in Figure 16. It is important to mention that the factor ϕ is independent of the injection velocity. Above viscosity ratios of 343, this equation cannot be used as the experimental M_e does not change with viscosity ratio. The difference in the fitting factor ϕ between the current experiments and the fit to data of Blackwell et al.²² could be because of the difference in the absolute permeability of the flow medium. The experiments of Blackwell et al.²² were conducted in sand packs of sand sizes as small as 100/140 mesh. The permeability of the parallel place cell with 1 mm spacing between the walls is approximately higher by a factor of 10 000.

Figures 17 shows the fingertip/front velocities for water injected in all the glycerol solutions at the three injection rates. The error bars on the data points represent the variation between two and three independent experimental runs. The front velocities follow a trend similar to the mixing zone velocities. It is observed that front velocities increase with the viscosity ratio up to a value of 343, above which the velocities plateau. Figure 18 shows the relative front velocity, defined as the ratio of front velocity to injection velocity (injection rate/cross section area) as a function of viscosity ratio at three injection rates. It is observed that the three curves converge, similar to observation for the mixing zone velocities.

In this paper we present an experimental study to investigate the growth of viscous fingers and mixing zone in rectilinear flow in Newtonian fluids. Glycerol solutions with viscosities covering the entire range from 1 to 1225 cP are displaced with water at different flow rates. The main conclusions are as follows.

• A systematic change in the viscous fingering patterns is observed with change in displaced fluid viscosity. At low viscosity ratios, the shielding effect is less pronounced and multiple fingers grow parallel to each other. At higher viscosity ratios, mechanisms including finger merging, coalescence, and shielding are prominent.

• Tip splitting is observed to be more pronounced at higher viscosity ratios. At low viscosity ratios, only one instance of tip splitting was observed. At moderate (100-300) and higher (300-1225) viscosity ratios, all the dominant fingers exhibit uneven tip splitting. At moderate viscosity ratios, the finger are observed to split into two fingers whereas at higher viscosity ratios, tip splitting into three or four fingers is also observed.

• All the experiments are performed at high Peclet numbers and a linear growth in the mixing zone length is observed in all the experiments. The mixing zone velocity is first observed to increase with the viscosity ratio and then reaches a constant value at high viscosity ratios (above 343). This observation is not captured by the Koval’s model, which was derived using experimental data for viscosity ratios below 150.

• Plots of dimensionless mixing zone velocity versus viscosity ratio at different flow rates converge into a single curve indicating a universal relationship between these quantities.

• The fingertip/front velocities also increase with viscosity ratio up to a viscosity ratio of 343 above which the velocities plateau.

The authors would like to acknowledge the financial support provided by RPSEA, the Department of Energy and the companies sponsoring the JIP on Hydraulic Fracturing and Sand Control at The University of Texas at Austin.