Viscoelastic Suppression of Gravity-Driven Counterflow Instability

Attempts to achieve"top kill"of actively flowing oil wells by insertion of dense drilling"muds", i.e., slurries of dense minerals, from above will fail if the Kelvin-Helmholtz instability in the gravity-driven counterflow produces turbulence that breaks up the denser fluid into small droplets. Here we estimate the droplet size to be sub-mm for fast flows and suggest the addition of a shear-thickening polymer to suppress turbulence. Laboratory experiments show a progression from droplet formation to complete turbulence suppression at the relevant high velocities, illustrating rich new physics accessible by using a shear-thickening liquid in gravity driven counter-streaming flows.

Dense fluids, i.e., mineral suspensions called "mud" [1,2], are introduced into oil wells to provide hydrostatic pressure to offset hydrocarbon (oil and gas) fluid pressure in deep formations, stopping upward flow and reducing the fluid pressure at the surface to near ambient. In a procedure known as "top kill" these muds are introduced at the top of the well. After pressure-driven injection they descend, filling it, in a gravity driven flow.
If hydrocarbon is flowing upward in the well there is a counterflow between the upwelling hydrocarbon and the descending mud. Successful top-kill requires that the mud descend despite this counterflow. However, upwelling at speeds 1 m/s, as in the uncontrolled Macondo well "blow-out" in the Gulf of Mexico in 2010, may lead to a Kelvin-Helmholtz instability [3]. In an attempt at top-kill this may disrupt the coherent downward stream of the mud by breaking it into small packets or droplets, and may explain the failure of the attempted top-kill of Macondo.
In order for a denser fluid to descend through an upwelling vertical column of lighter fluid it must not break up into packets or droplets whose settling velocity is less than the upwelling velocity. In vertical counterflows at high Reynolds number the Kelvin-Helmholtz instability grows and progresses rapidly to turbulence. This is in contrast to the case of horizontal flows in the stratified atmosphere or ocean, in which gravity and the vertical density gradient act to stabilize the instability.
If instability is not suppressed, the dense fluid introduced at the top of the column will be dispersed into small droplets that are spat out with the upwelling flow. In a "blown-out" oil well the consequence would be the failure of the dense fluid to accumulate to a depth sufficient to provide the hydrostatic pressure head necessary to "kill" (suppress the entry of hydrocarbon into) the well.
The purpose of this paper is to report experiments demonstrating that a surrogate mud is not dispersed by oil-mud counterflow when the mud consists of a shear-thickening viscoelastic suspension. Two-phase flows exhibit many complex phenomena [4,5]. There appear to be no previous studies of two-phase flows in which one of the fluids is viscoelastic.
Conventional drilling muds are typically Bingham plastics that flow as shear-thinning (pseudo-plastic) fluids above a small elastic yield stress [1,2,6,7]. This permits the mud to remove cuttings and debris and to keep them suspended during interruptions in drilling, while minimizing the required pumping pressure and power. In the following, we show that such fluids are likely to be dispersed into small pockets or droplets and carried out of the well bore by a rapid counterflow.
Because the theoretical estimates of this paper imply that instability and turbulent mixing will be severe with conventional muds, we suggest that the use of a shear-thickening mud that becomes elastic at high strain rates may enable top-kill of a rapidly flowing blownout well. We demonstrate in experiments with viscoelastic shear-thickening fluids complete suppression of the Kelvin-Helmholtz instability at flow velocities and Reynolds numbers approaching those of the blown-out Macondo well.
At lower flow rates thin filaments of immiscible fluid break up into droplets (the Plateau-Rayleigh instability [8]) under the influence of interfacial tension. We also demonstrate suppression of this instability in thin jets of viscoelastic fluid as elasticity prevents the rupture of the jet. In its place, we find novel "globule and filament" phenomenology.
Nonlinear instabilities and turbulence present formidable problems. We estimate the size of the droplets produced by turbulence, if it occurs, by assuming a Kolmogorov turbulent cascade [9] driven by the Kelvin-Helmholtz instability at an outer scale L and velocity U . L is taken as the diameter of the descending mud column (a fraction, determined by the mud flow rate, of the well bore diameter). U is the velocity difference between oil and mud, and must exceed the upwelling velocity of the oil for the mud to descend at all.
In an oil well L and U vary with depth. Turbulence is most intense and most effectively disperses the dense mud at the bottom of the well where L is smallest and U greatest, but these parameters do not vary by large factors, in part because in the upper, larger diameter, portions of the bore only an annulus may need to be filled [11].
Adopting U = 3.7 m/s (corresponding to Macondo's estimated [10] uncontrolled flow of 50,000 barrels/day (92 l/s) through the 0.18 m diameter bore at the bottom of the well [11]) and L = 0.09 m (considering a mud column half the bore diameter), and taking a representative kinematic viscosity of crude oil of 10 −5 m 2 /s [12] leads to a Reynolds number of roughly 40,000. Turbulence would be expected, but a well developed inertial subrange [9] would not be. The problem is complicated by the presence of two dynamically interacting fluids with very different properties. For lack of a better theory, we use the Kolmogorov spectrum in order to estimate the size of the resulting spatial structure.
If the fluids are miscible, the inner Kolmogorov scale (the smallest length scale on which the fluid is turbulent) [9] provides an estimate of the scale of compositional heterogeneity: where ≈ U 3 /L is the specific energy dissipation rate and ν is the kinematic viscosity.
The shear rate is high for r > r visc , but decays ∝ exp (−1.5(r/r visc ) −4/3 ) for r < r visc [9]. Structure on the scale r visc develops in a time L/U , but no finer spatial structure is expected. For miscible fluids the spatial structure does not take the form of spherical droplets because there is no interfacial energy acting to minimize the area of the boundary; we refer instead to "packets" of denser fluid.
The formation of a packet or droplet of one fluid surrounded by the other requires shear flow on the heterogeneity scale in both fluids. We adopt for the viscosity ν that of crude oil. For a comparatively inviscid, i.e., not shear thickening, mud, the characteristic size of the heterogeneity is r visc ∼ 0.05 mm. These denser mud volumes have a Stokesian descent (settling) velocity (for a density difference of 1 gm/cm 3 ) v descent < 1 cm/s, far less than the upward velocity of the fluid in which they are immersed. Even at the inner Kolmogorov scale the Reynolds stress ρu 2 k ∼ ρ( /k) 2/3 ∼ ρ( ν) 1/2 far exceeds the Bingham yield stress 10 Pa of typical drilling muds [1] so that these muds behave essentially as shear-thinning (pseudoplastic) fluids.
If the fluids are immiscible, as is the case with water-based muds, then interfacial energy further limits the droplet size, although it does not change the Kolmogorov inner scale of the turbulence within the homogeneous fluid. Equating the turbulent kinetic energy ρu 2 k /2 on a wave number scale k in the volume of a droplet of radius r surf to its interfacial energy yields 4πr 3 this is equivalent to the condition that the Weber number We ≡ ρu 2 The oil-water interfacial energy σ ≈ 0.025 N/m. The characteristic size of the droplets produced r surf ∼ 0.7 mm. The actual droplet sizes will be the greater of r visc and r surf ; It is possible to suppress instability and thereby to avoid dispersion of the mud into small droplets and its sweep-up by the counterflowing oil by adding to it a dilatant polymer with shear-thickening and viscoelastic properties. Before instability could grow to an amplitude sufficient to disperse the mud, the mud would have become very viscous or even elastic in the counterflow shear layer where dispersion would otherwise occur. Kelvin-Helmholtz instability is slowed by viscosity [3], and can be stabilized by elasticity. If instability occurred, once its growth produced a sufficient shear rate the mud would have become viscoelastic, suppressing further growth and preventing its dispersal. The resulting two-phase flow is difficult to predict and likely to be complex [4,5]. One of the purposes of our experiments was to study these phenomena.
Viscoelastic behavior is observed for mass ratios ρ * of corn starch to water in the range 1.1-1.7 (within this range there is little dependence on ρ * ). At low shear rates they shear-thin from a viscosity of about 100 Pa-s at 0.02/s to about 10 Pa-s at 2-5/s. Above this shear rate they abruptly shear-thicken in a jamming transition to a dynamic viscosity η jam 1000 Pa-s. At these higher shear rates the emulsion also has an elastic response that is difficult to quantify, but that is manifested in the trampoline-like behavior of pools of corn starch-water emulsions.
The Kolmogorov inner scale turbulent shear rate /ν ≈ U 3/2 / √ Lν ∼ 8000/s would far exceed that required for shear thickening and elasticity, so that the addition of corn starch to the mud would be expected to prevent its dispersal. Even the outer scale shear rate U/L ∼ 40/s at the mud-crude oil boundary is sufficient to put the mud in the shearthickened viscoelastic regime and to suppress the Kelvin-Helmholtz instability.
A column of dense viscoelastic fluid is thus predicted to remain coherent, with its flow including the Atwood number At ≡ (ρ water − ρ oil )/(ρ water + ρ oil ), were comparable to those of drilling mud in a light crude oil; the predicted droplet radius r surf ≈ 0.7 mm. Unlike drilling muds, our fluids contained no surfactant, so most of these small globules rapidly coalesced once the turbulence decayed, and descended in our static oil column ( Fig. 1(a)).
We then repeated the experiment using a strongly viscoelastic aqueous suspension earlier experiment. The Kelvin-Helmholtz instability was suppressed ( Fig. 1(b)), and the denser liquid descended as a coherent slug.
An instability analogous to the viscous buckling instability [17][18][19][20] began at the leading edge of the flow, and led to buckling and clumping into a slug of low aspect ratio. The initial stage of this instability is visible in Fig. 1(b) between the 100 cm and 110 cm marks.
The viscous buckling instability occurs at low Re. In our experiments Re is large in the oil (but not well defined in the viscoelastic corn starch suspension). Instability is initiated by inertial forces at the leading edge of the descending slug rather than by viscous forces, and its growth is limited by the viscoelastic properties of the slug. Fragmentation of the descending column was only observed at flow speeds of 2.5 m/s, and produced slugs of width comparable to the initial column radius.
The descent speed in high Re flow of a spherical slug whose size is that of an oil well bore would be ∼ 2 m/s. A slug elongated vertically, as in Fig. 1(b), would descend faster. The Reynolds number in the oil is large enough that the drag force is described by a coefficient of turbulent skin friction C skin ≈ 0.01 [21], leading to a terminal descent velocity v descent ≈ ∆ρgr/(C skin ρ) ≈ 7 m/s for representative ∆ρ/ρ = 1 and r = 50 mm. This is fast enough to overcome the upwelling even in an unusually rapidly flowing well like Macondo. In our experiments thick and vertically elongated columns of dense viscoelastic fluid were affected by the inertial forces at their leading surfaces; much longer columns and greater fluid masses would be required to study this regime past the influence of the leading surface.
At much lower flow rates the denser aqueous viscoelastic fluid stretched into a thin straight vertical filament, cohering viscoelastically, and thinning as it was accelerated by gravity. It did not disperse into drops of smaller diameter. In some trials, as illustrated in Fig. 1(c), it remained straight and unbroken for 1 s, demonstrating suppression of the Plateau-Rayleigh instability whose characteristic growth time ≈ 3 ρr 3 col /σ ≈ 7-20 ms for the column radius r col ≈ 0.5-1 mm [8].
In order to study the transition between droplet formation (at ρ * = 0) and continuous vertical flow (ρ * ≥ 1.2), we varied the suspension mass ratios and flow rates. A number of complex phenomena were observed. For example, at ρ * = 1.2 and flow rates of 1-2 ml/s the leading portion of the filament, after a steady descent of more than 1 m, broke up into a complex flow of globules (≈ 0.5 cm in diameter) strung together by thin filamentary loops, as shown in Fig. 2. This appears to be the result of the combined effects of the unstable dependence of flow rate on filament diameter (because of viscous drag by the surrounding fluid, in analogy to the instability of resistive hydraulic flow in open channels [22]) at low Reynolds numbers, the viscous buckling instability [17], the Plateau-Rayleigh instability [8], and viscoelasticity that inhibits breaking of the filaments even when they are greatly stretched and thinned.
We conclude that the addition of dilatant polymers to make a viscoelastic suspension is effective in suppressing both counterflow and surface tension instabilities and may, making a Galilean transformation of reference frames, maintain the coherence of a body of descending denser fluid in an upwelling flow. The potential practical application of these phenomena is the use of viscoelastic dense mud to fill a flowing (blown-out) oil well by enabling dense slugs of mud to descent against the upwelling oil. The hydrostatic pressure at the bottom may then be increased until it prevents entry of additional oil from the reservoir. This terminates the upwelling and "kills" the well. The phenomena observed go beyond this obvious practical application and focus attention on the rich physics of two counter-streaming fluids, one of which is shear-thickening and viscoelastic, which has so far received little attention.
We thank P. Dimotakis and R. Garwin for discussions that were the origin of this experiment and R. Grober for collaboration in a preliminary experiment with miscible fluids.